# Definite Integration of Absolute Value Functions

How to solve definite integration of absolute value functions problems: example and its solution.

## Example

Draw the graph *y* = |*x*^{2} - 1|.

In [-1, 1],

the graph of *y* = *x*^{2} - 1 (dashed)

is below the *x*-axis.

So draw the image of this dashed line

under the reflection in *y* = 0.

(which is above the *x*-axis.)

Graphing absolute value functions - Example 3

So *y* = |*x*^{2} - 1| is

-(*x*^{2} - 1) in [-1, 1],*x*^{2} - 1 in [1, 2].

So change the given integral into two integrals:

∫_{-1}^{1} (-*x*^{2} + 1) *dx* + ∫_{1}^{2} (*x*^{2} - 1) *dx*.

See the first integral.

The interval of the integral is symmetric: [-1, 1].

And the terms in (-*x*^{2} + 1) are all even functions.

So ∫_{-1}^{1} (-*x*^{2} + 1) *dx* = 2 ∫_{0}^{1} (-*x*^{2} + 1) *dx*

Definite integration of odd and even functions

Solve the definite integrals.

Definite integration of polynomials