301 Integral Calculus Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Integral calculus sits at the heart of engineering mathematics. Whether you are computing the area under a stress-strain curve, finding the volume of a structural member, determining the work done against a variable force, or modeling the heat distribution across a wall, integration is the tool that connects rate of change to total quantity. For Filipino engineering licensure examinees, a solid command of integral calculus is not optional. The PRC board exams for civil, mechanical, electrical, electronics, and chemical engineering consistently draw a significant portion of their mathematics questions from integration theory and its applications.

What makes integral calculus challenging is not just the computation. It is the conceptual variety. A single exam set may ask you to evaluate an improper integral, compute a centroid using double integration, apply the shell method to find a volume of revolution, solve a separable differential equation, or use the trapezoidal rule for numerical approximation. Each of these tasks requires a different setup, a different formula, and sometimes a different way of thinking about the problem. Reviewees who have memorized a few formulas without building conceptual depth tend to struggle when the question is presented in an unfamiliar way.

This reference list compiles 301 terms and definitions drawn from every area of integral calculus tested in Philippine engineering licensure examinations. The terms cover the full spectrum: foundational concepts like the Riemann sum and the Fundamental Theorem of Calculus, all major integration techniques from substitution to partial fractions to trigonometric substitution, the complete range of geometric and physical applications, numerical integration methods, convergence tests for improper integrals, and advanced topics like the Laplace transform, multiple integrals, and special functions. Every definition is written with the board exam in mind, giving you not just the meaning but the context you need to apply the concept under exam conditions.

Work through this list systematically and pay special attention to the application terms. Board exam problems in integral calculus are almost always application problems, not pure computation exercises. Knowing the definition of arc length is not enough. You need to recognize what the problem is asking, set up the correct integral, and choose the right technique to evaluate it. The terms in this list are organized alphabetically to make them easy to use as a quick reference during your review. Return to this list often, and use it alongside worked examples and practice problems for maximum effectiveness.

The 301 Integral Calculus Terms and Definitions

1. Absolute Convergence

A property of an infinite series where the series formed by taking the absolute values of all its terms also converges. A series that converges absolutely also converges in the ordinary sense, but the reverse is not always true. This distinction matters when working with alternating series in integral applications.

2. Absolute Value Integration

The process of integrating a function that involves the absolute value of an expression. It requires splitting the integral at the points where the expression inside the absolute value changes sign, then integrating each piece separately using the appropriate sign.

3. Absolutely Integrable Function

A function whose absolute value is also integrable over a given interval or unbounded domain. Absolute integrability is a stronger condition than ordinary integrability and is important in the theory of Fourier transforms and Lebesgue integration used in advanced engineering analysis.

4. Acceleration

In the context of integral calculus, acceleration is the derivative of velocity and the second derivative of position. Integrating acceleration gives velocity, and integrating velocity gives position. These relationships are central to problems involving motion, which appear regularly in PRC board exams.

5. Accumulation Function

A function defined as the integral of another function from a fixed lower limit to a variable upper limit. It measures the accumulated quantity of the integrand up to a given point and is the central object studied in the Fundamental Theorem of Calculus.

6. Algebraic Substitution

A form of integration by substitution in which an algebraic expression, typically involving radicals or composite polynomials, is replaced by a simpler variable. This technique converts the integrand into a polynomial or rational function that can be integrated directly using standard formulas.

7. Antiderivative

A function whose derivative equals a given function. Finding the antiderivative is the core task of indefinite integration. Every continuous function has infinitely many antiderivatives, all differing by a constant. The antiderivative is also called the primitive function or the indefinite integral.

8. Approximate Integration

The use of numerical methods to estimate the value of a definite integral when an exact antiderivative cannot be found or is impractical to compute. Approximate integration methods include the trapezoidal rule, Simpson’s rule, and Gaussian quadrature, all of which appear in engineering board exam problems.

9. Arbitrary Constant

The constant of integration C that appears in every indefinite integral. It represents all possible vertical shifts of the antiderivative and accounts for the non-uniqueness of the antiderivative. The arbitrary constant is determined by applying an initial or boundary condition.

10. Arc Length

The total length of a curve between two points, computed by integrating the square root of the sum of one and the square of the derivative of the curve. The formula applies to smooth curves in the plane and is a standard application topic in engineering board exams.

11. Area Between Curves

The area of the region bounded by two curves, computed by integrating the difference between the upper and lower function over the interval where they enclose the region. Identifying which curve lies above the other is a critical step that board exam problems often test.

12. Area Element

An infinitesimally small piece of area used in setting up a double integral. In rectangular coordinates the area element is dx times dy. In polar coordinates it becomes r times dr times d-theta. Choosing the correct area element is fundamental to setting up multiple integrals correctly.

13. Area in Polar Coordinates

The area enclosed by a polar curve, computed using the formula involving the integral of half the square of the radius function with respect to the angle. Problems require careful identification of the limits of integration based on where the curve closes.

14. Area Under a Curve

The region between a function and the horizontal axis over a given interval, measured using the definite integral. When the function is positive, the integral gives the exact geometric area. When the function dips below the axis, the integral accounts for sign, so care is needed to compute actual area.

15. Average Value of a Function

The mean value of a function over a closed interval, computed by dividing the definite integral of the function over the interval by the length of the interval. The result gives the height of a rectangle whose area equals the area under the curve on that interval.

16. Bernoulli Differential Equation

A nonlinear first order differential equation that can be reduced to a linear equation by a suitable substitution. The solution involves integration and is important in fluid mechanics and population dynamics, both of which appear in engineering board exam applications.

17. Beta Function

A special function defined by a definite integral involving two parameters and expressed in terms of factorials or the Gamma function. It appears in advanced integration problems and probability applications. The Beta function is related to the Gamma function through a well-known identity.

18. Boundary Condition

A condition specifying the value of the solution or its derivative at the boundary of the domain, used to determine the constants of integration in the general solution of a differential equation. Boundary conditions differ from initial conditions in that they may be specified at two different points.

19. Bounded Function

A function that does not grow without limit on a given interval, meaning its values stay within some fixed range. Boundedness is a prerequisite for the existence of the Riemann integral. Unbounded functions can still be integrated in some cases using improper integrals.

20. Bounded Region

A region in the plane that is enclosed and does not extend to infinity. Integration problems involving bounded regions ask for the area, volume, or other quantities associated with the enclosed space. Identifying the boundaries is the first step in setting up the integral.

21. Calculus of Variations

A branch of mathematics concerned with finding functions that optimize definite integrals. It generalizes ordinary optimization from points to entire functions. The Euler-Lagrange equation, derived using integral calculus, gives the condition that the optimal function must satisfy.

22. Cauchy Integral Theorem

A fundamental result in complex analysis stating that the integral of a complex analytic function around a closed contour is zero. While beyond standard board exam coverage, it has implications for the evaluation of certain real definite integrals using contour integration techniques.

23. Cavalieri’s Principle

A theorem stating that two solids with equal cross-sectional areas at every height have equal volumes. It provides the geometric foundation for computing volumes of solids of revolution and irregular solids using the method of cross sections.

24. Center of Gravity

The point at which the entire weight of a body can be considered to act, computed using integrals of the mass distribution over the body. In two dimensions, it requires computing the first moments of the region with respect to both axes and dividing by the total mass or area.

25. Center of Mass

The weighted average position of all mass in a system or body, computed using integrals in the continuous case. For a planar lamina with uniform density, the center of mass coincides with the centroid. This quantity appears in statics and structural engineering board problems.

26. Centroid

The geometric center of a region or solid, found by dividing the first moment of the region with respect to each axis by the total area or volume. For uniform density objects, the centroid equals the center of mass. Standard board exam problems involve centroids of triangles, semicircles, and composite shapes.

27. Chain Rule in Reverse

The conceptual description of integration by substitution, since the chain rule for differentiation generates composite expressions that substitution then undoes. Recognizing when an integrand has the structure of the output of the chain rule is the key step in applying substitution successfully.

28. Change of Variables

A technique for simplifying an integral by substituting a new variable in place of the original one, adjusting both the integrand and the limits of integration accordingly. It is the formal name for the substitution method, and it also applies in double and triple integrals through the Jacobian.

29. Characteristic Equation

The algebraic equation obtained by assuming an exponential solution to a linear differential equation with constant coefficients. Its roots determine the form of the complementary function. Integration is used to verify solutions and to find particular solutions by variation of parameters.

30. Circular Disk

The geometric shape whose area is used in the disk method for computing volumes of solids of revolution. The area of a circular disk is pi times the square of the radius, and integrating this area along the axis of revolution gives the total volume of the solid.

31. Closed Form Antiderivative

An antiderivative expressible using a finite combination of elementary functions such as polynomials, exponentials, logarithms, and trigonometric functions. Not all continuous functions have closed form antiderivatives, which is why numerical integration and series methods are also essential tools.

32. Comparison Test for Integrals

A method for determining whether an improper integral converges or diverges by comparing the integrand with a simpler function whose convergence behavior is known. If the simpler function converges and bounds the given function from above, the given integral also converges.

33. Comparison Theorem for Integrals

A result that allows the convergence or divergence of one integral to be concluded from the known behavior of another integral with a simpler integrand. If a non-negative integrand is dominated by a convergent integrand, the dominated integral also converges.

34. Complementary Function

In the context of differential equations solved by integration, the complementary function is the general solution to the associated homogeneous equation. It forms part of the complete solution when combined with a particular solution.

35. Completeness of Integration

The idea that every continuous function on a closed interval has a definite integral, a result guaranteed by the integrability of continuous functions. This property underlies the theoretical reliability of all integration methods used in engineering mathematics.

36. Composite Integration

Integration applied to a function that is itself composed of simpler functions, often requiring substitution or chain rule considerations in reverse. Recognizing the composite structure is key to choosing the right integration technique.

37. Composite Simpson’s Rule

The application of Simpson’s rule repeatedly over multiple subintervals of the integration domain. The composite version achieves higher accuracy than a single application by using a finer partition. It is the standard form of Simpson’s rule used in engineering numerical analysis.

38. Composite Trapezoidal Rule

The application of the trapezoidal rule over multiple subintervals, summing the areas of many small trapezoids. The composite trapezoidal rule converges to the exact integral as the number of subintervals increases and is straightforward to implement.

39. Concavity and Integration

The relationship between the concavity of a function and the error in numerical integration methods. The trapezoidal rule overestimates when the function is concave down and underestimates when it is concave up. Simpson’s rule is exact for quadratic functions regardless of concavity.

40. Conditional Convergence

A property of a series or improper integral that converges in the ordinary sense but does not converge absolutely. Conditionally convergent series are more sensitive to rearrangement, which can change the sum. The alternating harmonic series is a classic example.

41. Conservative Vector Field

A vector field that is the gradient of some scalar potential function. Line integrals of conservative vector fields are path independent, and their values depend only on the endpoints. Determining whether a field is conservative requires checking the equality of mixed partial derivatives.

42. Constant of Integration

The arbitrary constant added to every indefinite integral to account for the family of antiderivatives. Represented by C, it reflects the fact that any constant added to a function does not change its derivative. The constant is determined when initial or boundary conditions are given.

43. Continuity and Integrability

The relationship between a function being continuous and being integrable. Every function that is continuous on a closed interval is also Riemann integrable on that interval. Discontinuous functions may or may not be integrable depending on the nature of the discontinuities.

44. Continuous Function Integrability

The theorem guaranteeing that every function continuous on a closed bounded interval is Riemann integrable on that interval. This result is the most important sufficient condition for integrability in engineering mathematics and justifies the use of integration for all smooth physical quantities.

45. Contour Integration

The evaluation of a complex line integral along a specified path in the complex plane. In advanced engineering, contour integration is used to evaluate real improper integrals that are otherwise difficult to compute, by applying Cauchy’s residue theorem.

46. Convergence of Improper Integrals

The condition under which an improper integral has a finite, well-defined value. Convergence is tested by replacing the infinite limit or the problematic point with a variable limit and taking the limit of the resulting expression. If the limit is finite, the integral converges.

47. Convergent Integral

An improper integral whose value approaches a finite limit as the limits of integration approach infinity or a point of discontinuity. Verifying convergence is required before claiming a finite value for any improper integral. The comparison and limit comparison tests are the primary tools for this determination.

48. Convolution Integral

An integral that expresses the overlap between two functions as one is shifted over the other. It is widely used in engineering applications involving linear systems, signal processing, and the solution of differential equations using Laplace transforms.

49. Coordinate Transformation

The replacement of one coordinate system by another, such as converting from rectangular to polar or from Cartesian to spherical coordinates, in order to simplify the region of integration or the integrand. The Jacobian accounts for the change in the area or volume element.

50. Cumulative Distribution Function

In probability, the function giving the probability that a random variable is less than or equal to a given value, computed as the integral of the probability density function from negative infinity to that value. Integration is the fundamental operation connecting density to probability.

51. Curve Fitting and Integration

The process of fitting a curve to data points and then integrating the fitted function to estimate the area or accumulated quantity. This is a practical engineering skill combining numerical methods with integral calculus.

52. Cylindrical Shell Method

A technique for computing the volume of a solid of revolution by summing up thin cylindrical shells whose radii, heights, and thicknesses are expressed in terms of the variable of integration. It is particularly useful when the disk method produces complicated integrands.

53. Decomposition of Rational Functions

The process of expressing a rational function as a sum of simpler fractions before integration. This is the core operation in the method of partial fractions. Correct decomposition depends on correctly identifying the nature and multiplicity of the denominator’s factors.

54. Definite Integral

An integral evaluated between two specific limits, producing a single numerical value. Geometrically, it represents the signed area between the function and the horizontal axis over the given interval. The definite integral is the central object in applications of integral calculus.

55. Definite Integral as Area

The interpretation of the definite integral as the net signed area between the graph of a function and the horizontal axis. Regions above the axis contribute positive area and regions below contribute negative area. To find total unsigned area, split the integral at the zeros of the function.

56. Definite Integral as Limit of Sum

The formal definition of the definite integral as the limit of a Riemann sum as the width of the subintervals approaches zero. This definition connects the geometric concept of area with the algebraic process of summation and provides the theoretical basis for all numerical integration methods.

57. Definite Integral Properties

The set of algebraic and geometric rules governing the behavior of definite integrals. Key properties include linearity, additivity over intervals, sign reversal when limits are swapped, and the zero integral over a single point. These properties are used constantly in simplifying integral expressions.

58. Density Function

A function that describes how a quantity such as mass or charge is distributed over a region. Integrating the density function over a region gives the total amount of that quantity. Density functions appear in physics and engineering problems involving mass, pressure, and electrical charge.

59. Differential

An infinitesimally small change in a variable, denoted by dx or dy. In integration, the differential indicates the variable of integration and represents the width of an infinitely thin strip in the Riemann sum interpretation. Understanding differentials is essential for setting up integrals correctly.

60. Differential Equation

An equation that involves a function and one or more of its derivatives. Integral calculus provides methods for solving differential equations by integrating both sides, using separation of variables, or applying integrating factors. Many engineering problems reduce to solving differential equations.

61. Differential Form

An expression of the form f times dx plus g times dy representing a quantity to be integrated along a path. Recognizing whether a differential form is exact determines whether a potential function exists and whether the integral is path independent.

62. Disk Method

A technique for computing the volume of a solid of revolution by summing up thin circular disks perpendicular to the axis of revolution. The volume of each disk is pi times the square of the radius times the thickness, and the total volume is the integral of this expression.

63. Displacement vs Distance

The distinction between integrating velocity to get displacement, which accounts for sign, and integrating the absolute value of velocity to get total distance traveled. Board exam problems sometimes ask for one when the other seems more natural, so reading carefully is essential.

64. Distribution of Mass

The description of how mass is spread over a region, expressed as a density function. Integrating the density function over the region gives the total mass, and dividing the first moment by the total mass gives the center of mass.

65. Divergence of Improper Integrals

The condition in which an improper integral does not have a finite value. Divergence occurs when the limit defining the improper integral grows without bound or oscillates without settling. Recognizing divergence is as important as computing convergent integrals in exam settings.

66. Divergent Series

An infinite series whose partial sums do not approach a finite limit. Many tests, including the integral test and the comparison test, connect the divergence of series to the divergence of related improper integrals. Recognizing divergent series is as important as identifying convergent ones.

67. Double Angle Formulas in Integration

Trigonometric identities expressing sine and cosine of double angles in terms of single angle expressions, used to reduce even powers of trigonometric functions to integrable forms. These formulas are essential for integrating even powers of sine and cosine.

68. Double Integral

An integral taken over a two-dimensional region, computed by iterating two single integrals. The inner integral is evaluated first over one variable, and the result is integrated over the other variable. Double integrals are used to compute areas, volumes, and surface integrals.

69. Double Integration in Polar Form

The evaluation of a double integral after converting the region and integrand to polar coordinates. The area element becomes r times dr times d-theta. This approach simplifies integrals over circular or annular regions that are awkward to handle in rectangular form.

70. Dummy Variable

The variable of integration in a definite integral, which does not appear in the final result. It can be replaced by any other symbol without changing the value of the integral. Understanding dummy variables helps in recognizing equivalent integral expressions.

71. Elementary Function

A function built from a finite combination of polynomials, rational functions, exponentials, logarithms, and trigonometric functions using arithmetic operations and composition. Antiderivatives of elementary functions are not always themselves elementary, which motivates the study of special functions.

72. Elliptic Integral

A class of integrals that cannot be expressed in terms of elementary functions and arise in computing arc lengths of ellipses and periods of pendulums. They are defined by standard forms and are tabulated for numerical use. They appear occasionally in advanced engineering mathematics.

73. Energy and Integration

The use of integration to compute mechanical, thermal, or electrical energy from force, temperature, or voltage distributions. Energy is frequently expressed as the integral of power over time or force over displacement, making integration the central mathematical operation in energy calculations.

74. Error Function

A special function defined as the integral of the Gaussian function from zero to a given value, scaled by a constant. It arises in probability theory and the solution of heat conduction problems. Its values are typically obtained from tables or computed numerically.

75. Error in Numerical Integration

The difference between the exact value of a definite integral and its numerical approximation. Error bounds for the trapezoidal rule and Simpson’s rule depend on the second and fourth derivatives of the integrand respectively, and on the width of the subintervals used.

76. Euler’s Method

A numerical technique for approximating the solution of a differential equation by stepping forward in small increments using the slope of the function at each step. While not an integration technique in the classical sense, it relies on the same principles as numerical integration.

77. Evaluation Theorem

Another name for the second part of the Fundamental Theorem of Calculus, which states that the definite integral of a function can be computed by evaluating its antiderivative at the upper and lower limits and subtracting. This theorem makes definite integration computationally straightforward.

78. Even Function Integration

The property that the integral of an even function over a symmetric interval from negative a to a equals twice the integral from zero to a. This property significantly reduces computation when the integrand is recognized as even and the limits are symmetric.

79. Exact Differential

A differential expression that is the total derivative of some function, meaning it can be integrated directly to recover that function. Recognizing exact differentials is fundamental in solving exact differential equations and in line integral computations.

80. Exact Equation

A differential equation of the form M times dx plus N times dy equals zero where the expression M dx plus N dy is an exact differential. Recognizing an exact equation allows the solution to be found by integrating M with respect to x and adjusting for the y dependence.

81. Existence of the Integral

The conditions under which the definite integral of a function over a closed interval is guaranteed to exist. Continuity on the interval is sufficient but not necessary. Functions with finitely many jump discontinuities or bounded oscillations are also integrable in the Riemann sense.

82. Exponential Decay Integration

The integration of an exponential function with a negative exponent, producing another exponential function divided by the coefficient in the exponent. These integrals appear in problems involving radioactive decay, capacitor discharge, and population decline.

83. Exponential Growth and Decay

Mathematical models described by differential equations whose solutions involve exponential functions, obtained through integration with separation of variables. These models describe population growth, radioactive decay, and cooling processes. The constant in the solution is determined by initial conditions.

84. Extended Mean Value Theorem

A generalization of the Mean Value Theorem for integrals that involves a weight function. It states that the integral of the product of a function and a positive weight function equals the value of the function at some interior point times the integral of the weight function alone.

85. First Moment

The integral of the product of a function and the distance from a reference axis, used in computing centroids and centers of mass. The first moment of area with respect to an axis measures how far the area is distributed from that axis.

86. First Order Differential Equation

A differential equation involving only the first derivative of the unknown function. Many first order equations are solved by direct integration, separation of variables, or the use of integrating factors. They model a wide range of physical phenomena in engineering.

87. Fixed Limits of Integration

The specific values that define the bounds of a definite integral. The lower limit is written at the bottom of the integral sign and the upper limit at the top. Choosing correct limits is one of the most common sources of error in board exam problems.

88. Flux Integral

An integral that measures the flow of a vector field through a surface. It is computed as the dot product of the field with the outward normal to the surface, integrated over the surface area. Flux integrals appear in fluid dynamics and electromagnetics problems.

89. Flux Through a Surface

The rate at which a vector field passes through a surface, computed by integrating the normal component of the field over the surface area. Flux integrals are fundamental in fluid mechanics and electromagnetics and require setting up surface integrals with correct normal vectors.

90. Force and Pressure Integration

Applications of integration in computing the total force exerted by a fluid on a submerged surface, using the fact that pressure varies with depth. The integral sums the pressure contributions over the entire surface and yields the total hydrostatic force.

91. Fourier Series Integration

The use of integration to compute the coefficients of a Fourier series representation of a periodic function. The integrals exploit the orthogonality of sine and cosine functions over a full period to isolate each coefficient.

92. Fresnel Integral

A pair of special integrals involving the sine and cosine of the square of the variable, arising in optics and wave theory. They cannot be expressed in terms of elementary functions and are typically evaluated numerically or using tables.

93. Fubini’s Theorem

The theorem that guarantees the equality of the iterated integrals when integrating a continuous function over a rectangular or more general region. It justifies computing a double integral by integrating first with respect to one variable and then the other in either order.

94. Function Defined by an Integral

A function whose value at each point is defined as the definite integral of another function up to that point. The accumulation function is the primary example. Its derivative, given by the Fundamental Theorem of Calculus, equals the integrand evaluated at the variable upper limit.

95. Function of Bounded Variation

A function whose total variation over an interval is finite, meaning it does not oscillate infinitely. Functions of bounded variation are Riemann integrable, and this concept connects integrability with the behavior of the function across the interval.

96. Fundamental Theorem of Calculus, Part 1

The theorem stating that the derivative of the accumulation function of a continuous function equals the original function. It establishes that differentiation and integration are inverse operations and provides a rigorous foundation for all applied calculus.

97. Fundamental Theorem of Calculus, Part 2

The theorem stating that the definite integral of a continuous function over an interval equals the difference of its antiderivative evaluated at the upper and lower limits. This result transforms definite integration from a limiting process into a simple computation.

98. Gamma Function

A generalization of the factorial function to non-integer values, defined as an improper integral. The Gamma function satisfies the relation that it equals n factorial for positive integers. It appears in volume integrals, probability distributions, and advanced engineering analysis.

99. Gaussian Integral

The integral of the exponential of the negative square of the variable over the entire real line, which evaluates to the square root of pi. It is one of the most important integrals in probability and statistics and is computed using a clever squaring and polar coordinate technique.

100. Gaussian Quadrature

A numerical integration method that selects both the evaluation points and the weights optimally to achieve exact results for polynomials up to a specified degree with the fewest function evaluations. It is more accurate than Newton-Cotes methods for smooth integrands.

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101. General Solution

The complete family of solutions to a differential equation, expressed with one or more arbitrary constants depending on the order of the equation. Integration produces the general solution, and specific constants are found from initial or boundary conditions.

102. Generalized Power Rule

An extension of the power rule for integration to composite functions, where the result is the outer function raised to the appropriate power divided by the derivative of the inner function times the new exponent. This is essentially the substitution rule applied to a power function.

103. Geometric Applications of Integration

The collection of problems in which integration is used to compute geometric quantities such as area, arc length, surface area, and volume. These are the most common application problems on PRC engineering board exams and require correct setup of the integral as much as correct evaluation.

104. Geometric Series

An infinite series in which each term is a constant multiple of the previous term. Its sum can be computed using a closed formula when the common ratio has absolute value less than one. Geometric series arise in the context of power series representations of functions used in integration.

105. Gravitational Potential Energy

The energy stored in an object due to its position in a gravitational field, computed by integrating the gravitational force over the displacement. In engineering problems involving lifting loads or pumping fluids, the potential energy computation requires a carefully set up work integral.

106. Green’s Theorem

A theorem relating a line integral around a closed curve in the plane to a double integral over the region enclosed by the curve. It is a two-dimensional analog of the Fundamental Theorem of Calculus and is used to convert between line and area integrals.

107. Half-Angle Substitution

A substitution technique used to simplify integrals involving trigonometric functions by replacing trigonometric expressions with rational functions of the tangent of half the angle. It converts trigonometric integrands into rational functions that can be integrated using partial fractions.

108. Harmonic Series

The infinite series of reciprocals of the positive integers. The harmonic series diverges, meaning its partial sums grow without bound, albeit very slowly. This is a classic example used to illustrate the difference between convergence and divergence of series.

109. Heat Equation Solution by Integration

The process of applying integral transforms, particularly the Fourier or Laplace transform, to solve the partial differential equation governing heat conduction. Integration is central to both deriving and inverting these transforms to obtain the temperature distribution.

110. Heaviside Function

See Unit Step Function. A piecewise function defined as zero for negative arguments and one for positive arguments, used to model sudden switches in engineering systems. Its Laplace transform is one over s, and it is used to represent impulse inputs in circuit and control system analysis.

111. Higher-Order Integration

Integration repeated more than twice, producing triple, quadruple, or higher-order integrals. These are used in computing volumes and other quantities in three or more dimensions. The order of integration is chosen based on the limits and the shape of the region.

112. Homogeneous Differential Equation

A differential equation in which every term contains the dependent variable or its derivatives to the same degree. These equations are solved by substituting a ratio of the dependent and independent variables to reduce the equation to one that is separable.

113. Horizontal Integration

The practice of integrating with respect to y by treating the region as bounded by functions of y. This orientation is preferred when the boundaries of the region are more naturally described as horizontal strips, particularly when the region is bounded by curves that are more easily written as functions of y.

114. Hydrostatic Force

The total force exerted by a fluid on a submerged vertical surface, computed by integrating the product of fluid pressure and the width of the surface as a function of depth. Pressure increases linearly with depth, making the integral essential to the calculation.

115. Hyperbolic Function Integration

The integration of functions involving sinh, cosh, tanh, and their reciprocals. These integrals parallel the trigonometric integration formulas and arise in problems involving catenary curves, heat transfer, and electrical engineering applications.

116. Improper Integral Convergence Criteria

The set of conditions and tests used to determine whether an improper integral has a finite value. These include the comparison test, limit comparison test, Dirichlet’s test, and Abel’s test. Mastering these criteria is essential for dealing with integrals over infinite domains in engineering applications.

117. Improper Integral of the First Kind

An integral over an unbounded interval, where one or both limits of integration are infinite. It is evaluated by replacing the infinite limit with a finite variable, computing the definite integral, and taking the limit as the variable approaches infinity.

118. Improper Integral of the Second Kind

An integral of a function that has an infinite discontinuity at one or more points within the interval of integration. It is evaluated by splitting the integral at the discontinuity, replacing the point with a limit variable, and computing the resulting limit.

119. Indefinite Integral

The general antiderivative of a function, expressed with an arbitrary constant of integration. It represents the family of all functions whose derivative equals the given function. The indefinite integral is written using the integral sign without limits.

120. Indefinite Integration Rules

The complete set of standard antidifferentiation formulas used in computing indefinite integrals. These rules include the power rule, constant multiple rule, sum rule, and the specific formulas for exponential, logarithmic, and trigonometric functions. Fluency with these rules is the foundation of all integration work.

121. Initial Condition

A specific value of the dependent variable or its derivatives at a given point, used to determine the arbitrary constants in the general solution of a differential equation. Initial conditions convert the general solution into a particular solution.

122. Initial Value Problem

A differential equation paired with one or more initial conditions that specify the values of the solution and possibly its derivatives at a given starting point. Solving an initial value problem gives a unique particular solution.

123. Initial Value vs Boundary Value

The distinction between problems where the conditions on the solution are given at one point (initial value) and problems where conditions are given at two different points (boundary value). Both types require integration to solve, but boundary value problems may have multiple solutions or no solution.

124. Integrable Function

A function for which the definite integral exists and is finite over a given interval. Continuous functions are always integrable. Functions with a finite number of jump discontinuities or bounded discontinuities are also Riemann integrable.

125. Integral as Accumulation

The interpretation of the definite integral as the total accumulated amount of a quantity given its rate of change. This interpretation is the most physically intuitive way to understand integration and directly connects the mathematical operation to real-world engineering quantities like volume, mass, and energy.

126. Integral Bounds Switching

The rule that switching the upper and lower limits of a definite integral changes the sign of the result. This property is used frequently in manipulating integral expressions, particularly when combining or splitting integrals across subintervals.

127. Integral Calculus

The branch of calculus concerned with the theory and computation of integrals and their applications. Integral calculus encompasses indefinite integration, definite integration, techniques of integration, and the use of integrals to compute areas, volumes, arc lengths, and physical quantities.

128. Integral of a Composite Function

The integration of a function that is itself a composition of simpler functions. The key technique is substitution, which transforms the composite integrand into a simpler form. Recognizing the inner function and its derivative in the integrand is the critical skill.

129. Integral of a Constant

The result of integrating a constant function, which equals the constant times the variable of integration. This is the simplest integration formula and follows directly from the power rule for integration applied to the zero power.

130. Integral of Absolute Value Functions

The integration of a function involving absolute values, requiring the interval to be split at the zeros of the expression inside the absolute value. On each subinterval, the absolute value is replaced by the expression with the appropriate sign before integrating.

131. Integral of Exponential Functions

The integration formulas for functions of the form e raised to a power or a times x. The integral of e to the x is e to the x itself, and for e to the ax, the integral introduces a factor of one over a. These formulas are fundamental and appear constantly in board exam problems.

132. Integral of Hyperbolic Functions

The collection of integration formulas for sinh, cosh, tanh, and related functions. Each formula corresponds to a differentiation rule for hyperbolic functions applied in reverse. These integrals arise in structural and thermal engineering applications.

133. Integral of Inverse Trigonometric Functions

Integration formulas whose results are inverse trigonometric functions such as arcsin, arctan, and arcsec. Recognizing these forms in the integrand is a critical skill, as they appear frequently in engineering board exam problems involving radicals and quadratic expressions.

134. Integral of Logarithmic Functions

The integration of functions involving natural or common logarithms. The basic result that the integral of one over x equals the natural logarithm of the absolute value of x is one of the most frequently used formulas in all of calculus.

135. Integral of Power Functions

The power rule for integration, which states that the integral of x to the n equals x to the n plus one divided by n plus one, for any exponent except negative one. This rule is the most used integration formula in engineering mathematics.

136. Integral of Secant and Cosecant

The integration formulas for secant and cosecant, which produce logarithmic expressions involving combinations of trigonometric functions. These formulas are not immediately obvious from the functions themselves and are often remembered as special results to be memorized.

137. Integral of Trigonometric Functions

The standard integration formulas for sine, cosine, tangent, cotangent, secant, and cosecant. These formulas are derived from the corresponding differentiation rules and must be memorized for efficient performance on engineering board exams.

138. Integral Test for Series Convergence

A method for determining whether an infinite series converges by comparing it to a related improper integral. If the integral of the corresponding function converges, so does the series, and if the integral diverges, the series also diverges.

139. Integrand

The function being integrated in an integral expression. Identifying the integrand and understanding its structure is the first step in choosing the appropriate integration technique. The integrand appears between the integral sign and the differential.

140. Integration

The process of finding the antiderivative of a function or computing the definite integral between two limits. Integration is the inverse of differentiation and is used extensively in engineering to compute areas, volumes, work, and physical quantities.

141. Integration by Completing the Square

A technique for transforming a quadratic expression in the denominator or under a radical into a standard form that can be integrated using inverse trigonometric or logarithmic formulas. It is used when the integrand contains expressions like ax squared plus bx plus c.

142. Integration by Partial Fractions

A technique for integrating rational functions by decomposing them into simpler fractions whose integrals are known. The form of the decomposition depends on the nature of the factors in the denominator: linear, repeated, or irreducible quadratic.

143. Integration by Parts

A technique derived from the product rule for differentiation, expressed as the integral of u dv equals uv minus the integral of v du. It is used when the integrand is a product of two functions, and selecting u and dv correctly determines the success of the method.

144. Integration by Rationalization

A technique for integrating expressions involving radicals by substituting a variable that eliminates the radical, converting the integrand into a rational function. This method is particularly useful for integrands involving square roots of linear or quadratic expressions.

145. Integration by Substitution

The most fundamental integration technique, based on the chain rule in reverse. A portion of the integrand is replaced by a new variable u, the differential is adjusted accordingly, and the integral is evaluated in terms of u before converting back to the original variable.

146. Integration by Trigonometric Substitution

A technique for integrating expressions involving square roots of the form a squared minus x squared, x squared plus a squared, or x squared minus a squared, by substituting x equals a sine theta, a tangent theta, or a secant theta respectively to simplify the radical.

147. Integration Factor

A function multiplied through a linear first order differential equation to make the left side an exact derivative that can be integrated directly. The integrating factor is typically an exponential function of an integral of the coefficient of the dependent variable.

148. Integration in Economics

The application of integral calculus to economic concepts such as consumer surplus, producer surplus, and total revenue. These quantities are computed as definite integrals of demand or supply functions, connecting the mathematical tool to economic analysis encountered in engineering economics.

149. Integration of Discontinuous Functions

The process of integrating functions that have jump discontinuities or removable discontinuities. Riemann integration handles finitely many such discontinuities by treating them as isolated points that do not contribute to the integral. The integral is computed as normal, with the discontinuity having no effect on the result.

150. Integration of Piecewise Linear Functions

The integration of functions defined by different linear formulas on different subintervals. The total integral is computed as the sum of the integrals over each piece, each of which is the area of a trapezoid or triangle and can be computed geometrically as a check.

151. Integration of Rational Functions

The process of integrating a ratio of two polynomials. When the degree of the numerator is greater than or equal to the degree of the denominator, long division is performed first. The resulting proper fraction is then integrated using partial fractions.

152. Integration of Vector-Valued Functions

The component-wise integration of a function whose values are vectors. Each component is integrated separately with respect to the parameter. This is used in computing displacement from velocity and velocity from acceleration in vector form.

153. Integration over a Rectangle

A double integral computed over a rectangular region in the plane, where the limits of integration for each variable are constants. This is the simplest type of double integral and is used to introduce the concept of iterated integration before treating more general regions.

154. Integration over General Regions

Double or triple integrals computed over regions whose boundaries are described by functions of the other variable. Setting up these integrals correctly requires determining the correct limits of integration for each variable based on the shape of the region.

155. Inverse Laplace Transform

The operation that recovers the original time-domain function from its Laplace transform. It is typically computed using a table of known transform pairs and properties such as linearity and the first and second shift theorems. Integration is involved in the formal contour integral definition.

156. Iterated Integral

A multiple integral computed by performing successive single integrations, one variable at a time. The inner integral is evaluated first with the outer variable held fixed, and then the outer integral is computed. Fubini’s theorem guarantees that the order can be swapped under suitable conditions.

157. Iterated Integration Order

The decision about which variable to integrate first in a multiple integral. Choosing the better order can transform an intractable integral into a straightforward one. Reversing the order requires redescribing the region of integration in terms of the new outer variable.

158. Jacobian

A determinant used in the change of variables for multiple integrals, accounting for the distortion of area or volume when transforming from one coordinate system to another. The Jacobian must be included when converting between rectangular, polar, cylindrical, or spherical coordinates.

159. Kernel

A function of two variables appearing inside an integral transform, such as the Laplace or Fourier transform, that determines how the original function is transformed. The kernel encodes the relationship between the input and output domains of the transform.

160. Kinetic Energy Integration

The computation of the kinetic energy of a distributed mass system, such as a rotating body, using integration. The kinetic energy depends on the moment of inertia, which is itself computed by integration. These calculations appear in mechanical engineering board exam problems.

161. L’Hopital’s Rule in Integration

The use of L’Hopital’s rule to evaluate limits that arise when computing improper integrals or verifying the convergence behavior of integrands near points of discontinuity. While L’Hopital’s rule is a differentiation tool, it frequently assists in the analysis of integration problems.

162. Laplace Transform

An integral transform that converts a function of time into a function of a complex frequency variable, defined as the integral of the function times an exponential decay factor. It is widely used in engineering to solve linear differential equations by converting them to algebraic equations.

163. Laplace Transform of Derivatives

The rule expressing the Laplace transform of the derivative of a function in terms of the transform of the function itself and its initial values. This rule is what makes the Laplace transform so effective for solving initial value problems in engineering.

164. Left Riemann Sum

An approximation of a definite integral obtained by evaluating the function at the left endpoint of each subinterval and summing the resulting rectangles. For increasing functions, the left Riemann sum underestimates the true integral.

165. LIATE Rule

A mnemonic for selecting u in integration by parts, listing the preferred order as Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. Choosing u from the earlier category in this list typically simplifies the resulting integral.

166. Limit Comparison Test

A method for determining convergence of improper integrals by comparing the integrand to a simpler function with known behavior, using the limit of their ratio as the variable approaches the critical point. If the limit is a positive finite number, both integrals behave the same way.

167. Limit of Integration

Either of the two values that bound a definite integral. The lower limit is the starting value and the upper limit is the ending value of the variable of integration. Correct determination of these limits is the most common source of setup errors in application problems.

168. Line Integral

An integral along a curve in the plane or space, where the integrand is evaluated at each point on the curve. Line integrals are used to compute work done by a force field along a path and appear in engineering problems involving fluid flow and electromagnetics.

169. Linear Combination of Integrals

The expression of a complex integral as a sum of simpler integrals multiplied by constants. This follows from the linearity of integration and allows complicated integrands to be broken into manageable parts, each of which can be integrated using a standard formula or technique.

170. Linear Differential Equation

A differential equation in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. These equations are solved using integrating factors for first order equations or by superposition methods for higher order equations.

171. Logarithmic Integration

The technique of integrating expressions whose antiderivative involves a logarithm, particularly when the integrand is a rational expression where the numerator is the derivative of the denominator. Recognizing this pattern is a key skill in integration.

172. Lower Limit of Integration

The starting value of the variable in a definite integral, written at the bottom of the integral sign. The lower limit is substituted into the antiderivative during evaluation and subtracted from the value at the upper limit.

173. Lower Riemann Sum

The approximation of a definite integral using the minimum value of the function on each subinterval. The lower Riemann sum always underestimates the true integral for a positive function. The integral is defined as the common limit of the upper and lower Riemann sums.

174. Maclaurin Series Integration

The technique of integrating a function by first expanding it as a Maclaurin series, then integrating the series term by term. This is useful when the function does not have a known antiderivative but has a convergent power series representation.

175. Mass by Integration

The computation of the total mass of an object with continuously varying density using integration. The mass is the integral of the density function over the region occupied by the object. In one dimension, density varies along the length; in two or three dimensions, it varies over area or volume.

176. Mean Value Theorem for Integrals

The theorem stating that for a continuous function on a closed interval, there exists at least one point in the interval where the function equals its average value over the interval. The theorem guarantees the existence of this point but does not specify its location.

177. Method of Slicing

A technique for computing the volume of a solid by integrating the areas of its cross sections taken perpendicular to a chosen axis. The cross sections do not need to be circular, making this method more general than the disk or washer methods.

178. Midpoint Rule

A numerical integration method that approximates the definite integral by evaluating the function at the midpoint of each subinterval and summing the resulting rectangles. The midpoint rule is generally more accurate than the left or right Riemann sum for the same number of subintervals.

179. Mixed Partial Derivatives in Integration

The occurrence of both partial derivatives with respect to different variables in the integrand of a multiple integral. The order of integration and the limits may depend on the relationship between these partial derivatives and the region of integration.

180. Moment of Inertia

A measure of the resistance of a body to rotational acceleration about a given axis, computed by integrating the square of the distance from each mass element to the axis over the entire body. It is a key quantity in structural and mechanical engineering design.

181. Multiple Integral

An integral taken successively with respect to two or more variables, used to compute quantities over two- or three-dimensional regions. Double integrals compute planar areas and volumes, while triple integrals compute volumes and mass distributions in three dimensions.

182. Net Area

The value of the definite integral of a function, interpreted as the algebraic sum of positive and negative areas. When a function crosses the horizontal axis, the net area is less than the total geometric area. Total area requires splitting the integral and taking absolute values.

183. Net Change Theorem

The statement that the definite integral of the rate of change of a quantity gives the net change in that quantity over the interval. This theorem is a direct consequence of the Fundamental Theorem of Calculus and underlies all application problems involving rates.

184. Newton-Cotes Formulas

A family of numerical integration methods that approximate the definite integral by fitting polynomial curves through equally spaced function values. The trapezoidal rule and Simpson’s rule are the most commonly used members of this family in engineering applications.

185. Newton-Leibniz Formula

Another name for the second part of the Fundamental Theorem of Calculus, highlighting the contributions of both Newton and Leibniz to the development of calculus. It states that the definite integral equals the difference of the antiderivative evaluated at the limits.

186. Numerical Integration

The collection of techniques for approximating the value of a definite integral using finite sums, particularly when the antiderivative is difficult or impossible to express in closed form. Methods include the trapezoidal rule, Simpson’s rule, and Gaussian quadrature.

187. Odd Function Integration

The property that the integral of an odd function over a symmetric interval from negative a to a equals zero. This result follows from the cancellation of equal positive and negative contributions on opposite sides of the origin. Recognizing odd functions saves computation on board exams.

188. One-Sided Limit in Integration

The approach to a point from one side only, used in evaluating improper integrals at endpoints of discontinuity. The integral is split at the problematic point, and the limit is taken from the interior of the interval toward that point.

189. Order of Integration

The sequence in which the integrations in a multiple integral are performed. Changing the order of integration, when permitted by Fubini’s theorem, can simplify the computation significantly. Reversing the order requires adjusting the limits to describe the same region.

190. Orthogonality of Functions

The property of two functions whose integral of their product over a specified interval equals zero. Orthogonality is the basis for Fourier series and other eigenfunction expansions used extensively in engineering mathematics and physics.

191. Pappus’s Theorem on Surface Area

The theorem stating that the surface area of a solid of revolution equals the product of the arc length of the generating curve and the distance traveled by the centroid of the curve. It provides an elegant shortcut for surface area computations.

192. Pappus’s Theorem on Volume

The theorem stating that the volume of a solid of revolution equals the product of the area of the generating region and the distance traveled by the centroid of the region. It connects centroids with volume computations and is useful in engineering design.

193. Parametric Integration

The computation of integrals for curves defined by parametric equations, where both x and y are expressed as functions of a parameter. Area, arc length, and surface area formulas take different forms under parametric representation.

194. Partial Differential Equation

A differential equation involving partial derivatives with respect to two or more independent variables. Integral calculus, particularly Fourier and Laplace transforms, provides systematic methods for solving partial differential equations that model heat, wave, and potential problems.

195. Partial Fraction Decomposition

The algebraic process of expressing a rational function as a sum of simpler fractions before integration. The form of each fraction depends on whether the denominator factors are distinct linear, repeated linear, or irreducible quadratic factors.

196. Partial Sum

The sum of the first n terms of an infinite series. Convergence of a series is defined in terms of the limit of partial sums. In integral calculus, partial sums arise naturally when approximating definite integrals with Riemann sums using a finite number of subintervals.

197. Particular Solution

A specific solution to a differential equation obtained by applying initial or boundary conditions to the general solution. The particular solution satisfies both the differential equation and the given conditions, producing a unique function.

198. Path Independence

A property of a line integral that holds when the value of the integral depends only on the endpoints of the path and not on the specific curve taken between them. Path independence is equivalent to the vector field being conservative.

199. Physical Applications of Integration

The broad class of engineering and physics problems solved using definite integrals. These include work, energy, force, pressure, center of mass, moment of inertia, fluid flow, and heat transfer. Mastery of physical applications is the primary goal of integral calculus study for engineering licensure examinees.

200. Piecewise Integration

The technique of integrating a piecewise-defined function by splitting the integral at the breakpoints of the function and evaluating each piece separately with its corresponding formula. This approach is required whenever the integrand changes its formula over the interval.

201. Polar Area Formula

The formula for computing the area enclosed by a polar curve using the integral of half the square of the radius as a function of the angle. Identifying the correct angular limits is crucial and often requires finding where the curve intersects itself or the origin.

202. Polar Coordinates in Integration

The use of the polar coordinate system, where position is described by radius and angle, to evaluate integrals over regions that are more naturally described in polar form. The conversion introduces the Jacobian factor r into the area element.

203. Polar Curve

A curve described by expressing the radius as a function of the angle in polar coordinates. Integration in polar form computes areas enclosed by polar curves and arc lengths along polar curves. Identifying the range of the angle for which the curve traces a closed region is essential for setting up the integral.

204. Power Rule for Integration

The rule stating that the integral of x raised to a power n equals x raised to n plus one, divided by n plus one, plus a constant. This rule applies for all values of n except negative one, where the integral is instead the natural logarithm. It is the most essential integration formula.

205. Power Series

An infinite series of the form a sum of coefficients times powers of x minus a center point. Power series represent functions within their radius of convergence and can be integrated term by term within that radius. They are used to find antiderivatives of functions without closed-form expressions.

206. Power Series Integration

The technique of integrating a function represented as an infinite power series by integrating each term individually. The resulting series converges within the same radius of convergence as the original series. This method is used to find antiderivatives that are not elementary functions.

207. Pressure and Force in Fluids

The application of integration to compute the total force exerted by a fluid at rest on a submerged surface. Since pressure varies with depth, the total force requires integrating pressure times the width of the surface as a function of depth over the full height of the surface.

208. Principal Value Integral

The Cauchy principal value of an improper integral that would otherwise diverge due to a symmetric singularity, computed by taking symmetric limits around the problematic point. It provides a finite value for integrals that do not converge in the ordinary sense.

209. Probability Density Function

A function used in probability theory whose integral over any interval gives the probability of a random variable falling in that interval. The total integral over the entire domain must equal one. Integration is the fundamental operation connecting probability density to probability.

210. Product of Functions Integration

The integration of an expression that is the product of two or more functions. When no obvious substitution works, integration by parts is the primary technique. The LIATE rule guides the choice of which factor to differentiate and which to integrate.

211. Product Rule Reversed

The conceptual basis for integration by parts, viewing the technique as the reverse application of the product rule for differentiation. The formula uv minus the integral of v du recovers the area contribution missed by a simple antiderivative when the integrand is a product.

212. p-Series Test

A test for the convergence of the infinite series of one over n raised to the power p. The series converges when p is greater than one and diverges when p is one or less. This result is connected to the behavior of related improper integrals through the integral test.

213. Pumping Work

The work required to pump a fluid out of a container, computed by integrating the weight of each horizontal layer of fluid times the distance it must be lifted. The total work is the integral of this product over the full depth of fluid, a classic application in civil and mechanical engineering board exams.

214. Quadrature

The classical term for the numerical or analytical computation of a definite integral. Gaussian quadrature is a sophisticated method that selects both the evaluation points and the weights optimally to achieve high accuracy with few function evaluations.

215. Radius of Convergence

The value that defines the interval within which a power series converges. When integrating a power series term by term, the resulting series has the same radius of convergence as the original. Outside this radius, the series and its integral both diverge.

216. Rate of Change and Integration

The principle that integrating a rate of change over an interval gives the total change in the quantity. This is the core idea behind the Net Change Theorem and underlies all applications of integral calculus to physics, engineering, and economics.

217. Ratio of Functions Integration

The integration of an expression that is a ratio of two functions. When the numerator is the derivative of the denominator, the result is a logarithm. When the ratio is a proper rational function, partial fractions apply. Recognizing the type of ratio determines the correct approach.

218. Ratio Test

A test for the convergence of infinite series based on the ratio of successive terms. When the limiting ratio is less than one, the series converges absolutely. It is applied in determining the convergence of power series used in integration.

219. Rectangular Approximation Method

The use of rectangles to approximate the area under a curve, which is the intuitive foundation of Riemann integration. Left, right, and midpoint rectangles give different approximations, and the definite integral is the common limiting value as the rectangles become infinitely thin.

220. Recurrence Relation

A formula expressing the integral of a function with a certain index in terms of an integral with a lower index. This is the precise definition of a reduction formula. Recurrence relations allow high-power integrals to be reduced step by step to simpler base cases.

221. Recursive Integration

The repeated application of an integration technique, such as integration by parts, to the same integral type until a base case is reached. This is how reduction formulas are applied in practice. The process terminates when the remaining integral is one of the standard forms.

222. Reduction Formula

A formula that expresses the integral of a power of a function in terms of the integral of a lower power of the same function. Reduction formulas are derived by applying integration by parts repeatedly and are used to integrate powers of trigonometric and exponential functions.

223. Region of Integration

The set of points over which a multiple integral is evaluated. Correctly identifying and describing the region of integration, including its boundaries and the order of the variables, is the most critical step in setting up double and triple integrals.

224. Reversing the Order of Integration

The technique of changing the order in which the two variables are integrated in a double integral, requiring the limits to be rewritten to describe the same region in the new order. This can simplify an integral that is difficult or impossible to evaluate in the original order.

225. Riemann Integral

The classical formulation of the definite integral as the limit of Riemann sums. A function is Riemann integrable on an interval if and only if the upper and lower Riemann sums converge to the same limit as the partition is refined. Most functions encountered in engineering are Riemann integrable.

226. Riemann Sum

An approximation of a definite integral formed by partitioning the interval into subintervals, evaluating the function at a sample point in each subinterval, multiplying by the width, and summing the results. The definite integral is the limit of Riemann sums as the partition width approaches zero.

227. Right Riemann Sum

An approximation of a definite integral obtained by evaluating the function at the right endpoint of each subinterval. For increasing functions, the right Riemann sum overestimates the true integral, while for decreasing functions it underestimates.

228. Rolle’s Theorem Application in Integration

The use of Rolle’s Theorem to locate zeros of the derivative of the antiderivative, which corresponds to extreme values of the antiderivative itself. This connection between Rolle’s Theorem and the geometry of the antiderivative is used in verifying integration results.

229. Root Test

A convergence test for infinite series based on the nth root of the absolute value of the nth term. When this limit is less than one, the series converges absolutely. The root test is particularly effective for series involving nth powers.

230. Rotation About the x-Axis

The generation of a solid of revolution by rotating a region in the plane about the horizontal axis. The volume is computed using the disk or washer method, integrating with respect to x. The choice between disk and washer depends on whether there is a hole in the solid.

231. Rotation About the y-Axis

The generation of a solid of revolution by rotating a region in the plane about the vertical axis. The volume can be computed using the disk or washer method integrated with respect to y, or using the cylindrical shell method integrated with respect to x.

232. Scalar Line Integral

A line integral in which the integrand is a scalar function evaluated along the curve, multiplied by the arc length element. It computes the total accumulated value of the scalar quantity along the path, such as the total mass of a wire with variable linear density.

233. Second Moment

The integral of the square of the distance from a reference axis weighted by the mass or area element, giving the moment of inertia or second moment of area. It measures the distribution of area or mass relative to an axis and appears in bending stress and deflection calculations in civil engineering.

234. Separable Differential Equation

A differential equation that can be written with all terms involving the dependent variable on one side and all terms involving the independent variable on the other side, allowing both sides to be integrated separately. Separation of variables is the simplest technique for solving differential equations.

235. Separation of Variables

The technique of solving a differential equation by algebraically separating the variables and integrating both sides independently. It applies to equations where the right-hand side can be factored as a product of a function of x and a function of y.

236. Separation of Variables in PDEs

The technique of assuming that the solution of a partial differential equation is a product of functions each depending on only one variable, then integrating or solving each resulting ordinary differential equation separately. It is the standard method for solving the heat, wave, and Laplace equations.

237. Series Expansion in Integration

The technique of representing a function as an infinite series and integrating the series term by term. This approach is used when the function does not have a closed-form antiderivative, yielding the antiderivative as an infinite series within the radius of convergence.

238. Simpson’s Rule

A numerical integration method that approximates the definite integral by fitting quadratic polynomials through groups of three equally spaced points and summing the resulting areas. Simpson’s rule is more accurate than the trapezoidal rule and is widely used in engineering numerical analysis.

239. Simpson’s Three-Eighths Rule

A variant of Simpson’s rule that uses cubic polynomial approximation over four equally spaced points. It provides higher accuracy than the standard Simpson’s rule for the same interval width and is used when even greater precision is required in numerical integration.

240. Single Variable Integration

Integration with respect to one variable, producing a function of the remaining variables or a constant when evaluated as a definite integral. Single variable integration is the foundation upon which multiple integration is built through the iterated integral approach.

241. Smooth Curve

A curve along which the derivative exists and is continuous, ensuring that the arc length integral is well-defined and finite. Smooth curves are the primary setting for arc length, surface area, and line integral computations in engineering mathematics.

242. Solid of Revolution

A three-dimensional solid formed by rotating a two-dimensional region about a line in the plane. The volume of a solid of revolution is computed using the disk, washer, or shell method depending on the orientation and the presence of holes.

243. Special Integrals

A collection of integrals that do not have elementary antiderivatives but arise frequently in engineering and physics. Examples include the Gaussian integral, elliptic integrals, the error function, and Fresnel integrals. Their values are typically obtained from tables or numerical computation.

244. Spherical Coordinates in Integration

A coordinate system using radius, polar angle, and azimuthal angle to describe points in three-dimensional space. Triple integrals in spherical coordinates include the Jacobian factor of r squared times sine of the polar angle. This system is most effective for regions with spherical symmetry.

245. Spring Work

The work done in compressing or stretching a spring, computed using integration because the force varies with displacement according to Hooke’s Law. The integral of the force function over the displacement gives the total work, a standard application problem in engineering board exams.

246. Squeeze Theorem in Integration

The use of the squeeze theorem to evaluate limits that arise in improper integrals or in verifying the convergence of integrals by bounding the integrand between two functions with known integrals.

247. Standard Form of Integrals

The collection of basic integral formulas that serve as the building blocks for all integration. These include the power rule, logarithmic form, exponential form, and the six trigonometric integrals. Mastery of standard forms is the foundation of efficient integration on board exams.

248. Standard Substitution Forms

A collection of substitution patterns that arise frequently enough to be recognized on sight. Examples include substituting u equals x squared plus one for an integrand involving two x times f of x squared plus one, or substituting u equals sin x when cos x appears as a factor.

249. Step Function

A function that is constant on each of a set of intervals and has jump discontinuities between them. Step functions are integrable since their discontinuities form a set of measure zero. The integral of a step function is computed as a sum of rectangular areas.

250. Stokes’ Theorem

A generalization of Green’s Theorem to three dimensions, relating the surface integral of the curl of a vector field over a surface to the line integral of the field around the boundary of the surface. It is a fundamental result in vector calculus and electromagnetics.

251. Substitution Rule

The formal statement of integration by substitution, which says that the integral of a composite function times the derivative of the inner function equals the integral of the outer function with respect to the new variable. It is the integration analog of the chain rule.

252. Sum of Infinite Series by Integration

The technique of recognizing the partial sums of a power series as the result of integrating a known function and using that connection to find the sum of the series. This connects the theory of series with the theory of integration in a powerful way.

253. Sum Rule for Integration

The property that the integral of a sum of functions equals the sum of their integrals. This rule allows complex integrands to be broken into simpler parts that are integrated separately and is one of the foundational properties of the integral.

254. Superposition Principle

The property of linear systems and equations that allows complex solutions to be constructed as sums of simpler solutions. In the context of integral calculus, superposition means that the integral of a sum is the sum of the integrals, a direct consequence of the linearity of the integral.

255. Surface Area of Revolution

The area of the surface formed by rotating a curve about a line, computed using an integral involving the arc length element and the distance of the curve from the axis. The formula differs depending on whether the rotation is about the x-axis or the y-axis.

256. Surface Integral

An integral over a two-dimensional surface in three-dimensional space. It generalizes the concept of a line integral and is used to compute flux through a surface, the mass of a curved surface with variable density, and other physical quantities in engineering.

257. Table of Integrals

A reference list of common and special integrals presented without derivation. Engineers use tables of integrals to quickly look up results for standard forms that would take time to rederive. Knowing how to recognize when an integrand matches a tabulated form is a practical skill.

258. Tabular Integration

A systematic method for performing integration by parts multiple times, organized in a table format. The function to be differentiated is listed in one column and the function to be integrated in another, with signs alternating. It is efficient for products of a polynomial and an exponential or trigonometric function.

259. Tangent Line Approximation and Integration

The use of tangent line approximations to estimate integrands or to simplify integrals near a specific point. While less precise than the integral itself, tangent line approximations provide quick estimates useful in engineering contexts where order of magnitude answers suffice.

260. Taylor Series Integration

The technique of expanding a function as a Taylor series centered at a point and then integrating the series term by term. The result provides the antiderivative as a power series, which is particularly useful for functions with no closed-form antiderivative.

261. Telescoping Series

A series in which most terms cancel with adjacent terms, leaving only the first and last. The sum of a telescoping series is easy to compute and is often used in demonstrating convergence results related to integrals expressed as limits of sums.

262. Terminal Velocity

The constant velocity reached by a falling object when the drag force equals gravity. It is computed by solving a differential equation using integration with separation of variables. Terminal velocity problems are common applications in engineering board exams.

263. Torque and Integration

The computation of total torque or moment about an axis using integration when the force or distance varies continuously. It is the rotational analog of work and uses the same integral framework. Torque problems appear in mechanical and structural engineering contexts.

264. Total Area

The sum of all geometric areas bounded by a curve and the horizontal axis, regardless of sign. To compute total area, the integral is split at the zeros of the function and the absolute value is taken of each sub-integral before summing. Total area is always non-negative.

265. Total Change

The accumulated amount by which a quantity changes over an interval, computed as the definite integral of its rate of change. This is a direct application of the Net Change Theorem and underlies all engineering use of integration for cumulative quantities.

266. Total Work Done by a Variable Force

The total energy transferred by a force that varies with position, computed by integrating the force function over the displacement. When the force is constant, work equals force times distance, but when force varies, the integral accounts for the variation.

267. Transformation of Limits

The adjustment of the limits of integration when applying substitution to a definite integral. When the variable is changed from x to u, the limits must be transformed using the substitution formula so the integral is evaluated directly in terms of u without converting back.

268. Trapezoidal Rule

A numerical integration method that approximates the area under a curve by dividing the interval into subintervals and approximating each strip with a trapezoid. The method is simple and widely used, with accuracy improving as the number of subintervals increases.

269. Trigonometric Identities in Integration

The use of identities such as the Pythagorean identities, double angle formulas, and product-to-sum formulas to transform a trigonometric integrand into a more integrable form. Selecting the right identity is a strategic skill tested on engineering board exams.

270. Trigonometric Integration

The integration of functions composed of trigonometric expressions, which often requires the use of identities, substitution, or reduction formulas. Powers of sine and cosine are the most common forms, and the strategy depends on whether the powers are odd or even.

271. Trigonometric Power Integrals

Integrals involving powers of sine, cosine, tangent, and other trigonometric functions. The strategy depends on whether the powers are odd or even. Odd powers allow a factor to be reserved for the substitution differential, while even powers require the use of half-angle or double-angle identities.

272. Trigonometric Substitution

See Integration by Trigonometric Substitution. A technique that replaces algebraic expressions involving radicals with trigonometric expressions, simplifying the integral into a trigonometric form that can be evaluated using standard trigonometric integrals.

273. Triple Integral

An integral evaluated over a three-dimensional region by iterating three single integrations. Triple integrals compute volumes, masses, and moments of three-dimensional solids. The order of integration is chosen to match the boundaries of the region as described in the problem.

274. Triple Integral in Cylindrical Coordinates

A triple integral evaluated after converting the region to cylindrical coordinates, which use radius, angle, and height. The Jacobian factor r is introduced in the volume element. This system is most convenient for regions with cylindrical symmetry such as cylinders and cones.

275. Triple Integral in Spherical Coordinates

A triple integral evaluated in spherical coordinates using radius, polar angle, and azimuthal angle. The volume element includes the Jacobian r squared sine of the polar angle. This system is ideal for regions with spherical symmetry such as balls and shells.

276. u-Substitution

The informal name for integration by substitution, where the inner function is labeled u and the differential is expressed in terms of du. This relabeling simplifies the integral into a standard form. It is the most commonly applied technique in introductory integral calculus.

277. Unbounded Integrand

An integrand that becomes arbitrarily large near a point within or at the boundary of the interval of integration. Integrals with unbounded integrands are improper integrals of the second kind and must be evaluated using limits. They may converge to a finite value despite the integrand becoming infinite.

278. Uniform Convergence

A mode of convergence for a sequence of functions in which the convergence is equally rapid at all points in the domain. Under uniform convergence, integration and the limit operation can be interchanged, justifying the term-by-term integration of uniformly convergent series.

279. Uniform Partition

A partition of the interval of integration into subintervals of equal width, used in the standard definitions of Riemann sums and numerical integration methods. The common width of the subintervals is the total interval length divided by the number of subintervals.

280. Unit Step Function

A function that is zero for negative values and one for positive values, also called the Heaviside function. Its Laplace transform is used in engineering to model sudden changes in inputs to systems, and its integral equals the ramp function.

281. Upper Limit of Integration

The ending value of the variable in a definite integral, written at the top of the integral sign. The antiderivative is evaluated at the upper limit and the lower limit, and the difference gives the value of the definite integral.

282. Upper Riemann Sum

The approximation of a definite integral using the maximum value of the function on each subinterval. The upper Riemann sum always overestimates the true integral for a positive function. The definite integral is the limit of both upper and lower Riemann sums as the partition is refined.

283. Variable Density

A density function that changes from point to point within a region, requiring integration to compute total mass, center of mass, or moment of inertia. Variable density problems are common in civil and mechanical engineering and test the ability to correctly set up and evaluate mass integrals.

284. Variable Force

A force that changes in magnitude or direction as a function of position, requiring integration to compute the work done. The classic example is a spring obeying Hooke’s Law. Variable force problems appear frequently in mechanical and civil engineering board exams.

285. Variable of Integration

The symbol indicating which variable is being integrated, appearing in the differential at the end of the integrand. In single integrals this is typically x or t, and in multiple integrals additional variables are introduced for each level of integration.

286. Variable Upper Limit Integral

A definite integral in which the upper limit is a variable rather than a constant, producing a function of that variable. This is precisely the accumulation function studied in the Fundamental Theorem of Calculus. Differentiating a variable upper limit integral recovers the integrand.

287. Vector Field Integration

The integration of a vector-valued function over a curve or surface, yielding scalar quantities such as work and flux. Line integrals and surface integrals are the two main types. These tools are foundational in electromagnetics, fluid mechanics, and heat transfer in engineering.

288. Velocity and Displacement

The relationship between velocity and position exploited in integral calculus. Integrating velocity gives displacement, and integrating the absolute value of velocity gives total distance traveled. These computations are a staple of PRC engineering board exam problems.

289. Vertical Integration

The practice of integrating with respect to x by treating the region as bounded by vertical strips extending from the lower to the upper bounding curve. This is the standard approach for most area and volume problems and is natural when the boundaries are expressed as functions of x.

290. Volume by Cross Sections

The computation of the volume of a solid by integrating the area of its cross sections along an axis. The cross sections may be squares, semicircles, triangles, or other shapes. This method generalizes the disk method and applies to solids that are not solids of revolution.

291. Volume by Double Integration

The use of a double integral to compute the volume of a solid under a surface defined by a function of two variables. The function gives the height at each point, and integrating over the region in the base plane gives the total volume.

292. Volume Element

An infinitesimally small piece of volume used in setting up a triple integral. In rectangular coordinates the volume element is dx times dy times dz. In cylindrical coordinates it becomes r times dr times d-theta times dz. In spherical coordinates it is r squared times sine of phi times dr times d-theta times d-phi.

293. Volume of a Solid of Revolution

The volume of the three-dimensional body formed by rotating a two-dimensional region about an axis, computed using the disk, washer, or shell method. This is one of the most frequently tested topics in the integral calculus portion of Mathematics engineering board exams.

294. Wallis Formula

A product formula for pi derived from the integral of even and odd powers of cosine over a half period. The Wallis formula expresses pi as an infinite product involving even and odd integers and is a beautiful result connecting integration to the fundamental constant pi.

295. Wallis Integrals

The definite integrals of sine or cosine raised to integer powers over a half period from zero to pi over two. They are evaluated using reduction formulas and produce results involving products of odd or even integers depending on whether the power is odd or even.

296. Work Done Against Gravity

The energy required to lift an object from one height to another, computed by integrating the gravitational force over the displacement. When the mass is distributed continuously, as in lifting a cable or pumping a fluid, integration accounts for the varying height of each mass element.

297. Work Integral

The definite integral that computes the work done by a variable force acting along a displacement. Work equals the integral of force with respect to displacement over the path. When force and displacement are vectors, the line integral of force along the path gives the total work.

298. Wronskian

A determinant formed from a set of functions and their derivatives, used to test whether the functions are linearly independent. A non-zero Wronskian confirms that the functions form a fundamental set of solutions to a differential equation, which is key to writing the general solution.

299. Zero Integral Property

The property that the definite integral of any function over an interval of zero length equals zero. This follows directly from the definition of the definite integral and is used to justify collapsing integrals to zero when the limits of integration coincide.

300. Zero Net Change

The condition in which a quantity changes at a positive rate for part of an interval and at an equal negative rate for the remainder, resulting in a net integral of zero. This occurs for odd functions integrated over symmetric intervals and whenever equal positive and negative areas cancel.

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301. Zero of a Function and Integration

The points where a function equals zero are used to determine where the function changes sign, which is essential for computing the total geometric area between the function and the horizontal axis. These points split the integral into regions where the function is positive or negative.

Conclusion

Integral calculus is one of the widest subjects in the Philippine engineering board exam mathematics coverage, and this list of 301 terms reflects that breadth. As you prepare, remember that the board exam rewards conceptual understanding as much as computational skill. The Fundamental Theorem of Calculus, the standard integration formulas, integration by parts, substitution, partial fractions, and trigonometric substitution form the technical core. But the application topics: areas, volumes, arc length, work, centroids, and hydrostatic force are where most of the actual exam points are earned. Make sure you can set up these integrals from a word problem, not just evaluate them when the integral is already written out for you.

Give extra attention to the topics that appear most frequently across different engineering disciplines. The disk and shell methods for volumes of revolution are universal. The work integral with a variable force appears in both mechanical and civil engineering contexts. Separable differential equations and their solutions by integration appear across all disciplines. Improper integrals and convergence tests become more important as you advance to electronics and electrical engineering topics. And the Laplace transform, which is built entirely on integral calculus, is indispensable for any examinee taking the electrical and electronics engineering board exams.

Finally, do not underestimate the numerical integration methods. The trapezoidal rule and Simpson’s rule appear regularly in engineering board exam problems, especially when the integrand is given as a table of values rather than a formula. Knowing when and how to apply these methods, and understanding their error behavior, distinguishes a well-prepared examinee from one who has only studied the analytical techniques. Integral calculus rewards systematic preparation. Study the definitions, practice the techniques, and drill the applications. The board exam is very much a test of applied competence, and this list gives you the vocabulary to build that competence on solid ground.

For practice problems on all these topics, head over to our Integral Calculus Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.