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Lecture in Definite Integrals

(Last Updated On: March 2, 2019)
Lecture on Definite Integral
Definite integrals are used to find the area between the graph of a function and the x-axis. If a continuous function f(x) is positive over the domain , then the area under its graph is

Lecture in Definite Integrals

Where:

  • F(x) is the integral of f(x);
  • F(b) is the value of the integral at the upper limit, x=b; and
  • F(a) is the value of the integral at the lower limit, x=a.

Note: It does not involve a constant of integration (arbitrary constant) and it gives us a definite value (a number) at the end of the calculation.

Properties of the Definite Integral:

  • Integral of a constant:

    (a)

    (b) Lecture in Definite Integrals

  • Linearity:

    (a) Lecture in Definite Integrals

    (b) Lecture in Definite Integrals if c is a constant.

  • Interval Additivity

    (a) Lecture in Definite Integrals

    (b) Lecture in Definite Integrals

    (c) Lecture in Definite Integrals If a < b then it is convenient to define

  • Comparison:

    Lecture in Definite Integrals If 0 < f(x) < g(x) for all x in [a, b],

The Evaluation Theorem

If is a continuous function and F is an antiderivative of f,   F'(x) = f(x), then

 

 Example: Using the Theorem, Find the value of  .

  • Applying what you learned, an antiderivative of x2 is
  • Then, evaluate and substitute the limit. 
  • = with limit 0 to 1 =  

At this point, you are ready to answer some problems involving definite integral. Follow the link to start. Definite Integral – Set 1 Problems

If you have some clarifications. Let me know.

credit: Renato E. Apa-ap, et al.

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