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 isWhere:

- 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.**Interval Additivity**(a)

(b)

(c) If

**a < b**then it is convenient to define**Comparison:**If

**0**for all__<__f(x)__<__g(x)**x**in**[a, b]**,

### The Evaluation Theorem

If *f *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
*x*^{2}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.*