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.

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.