# Lecture in Definite Integrals

(Last Updated On: March 2, 2019) 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 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) • Linearity:

(a) (b) if c is a constant.

(a) (b) (c) If a < b then it is convenient to define

• Comparison: 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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