Definite Integration Involving Modulus Function

There is no anti-derivative for a modulus function; however we know it’s definition

Thus we can split up our integral in two parts. One part must be completely negative and the another must be completely positive. This strategy is known as splitting . To split correctly it is better to draw the modulus graph so that we can get the correct equation to represent each portion of the modulus graph .

**Example 1**. ** Find the value of **

**
.**

**Sol. ** For a periodic function with period , , where is any natural number.

Here whose period is .

Hence

= ( since )

= = 9 x 2 =18 .

Thus

**Example 2. Evaluate **

**Sol. **We note that on and on and that on .

**Graph of **

So

=

=

=

=

**Example 3. Find the value **

**Sol. ** We know that and has one common value at . In the interval , and in , . Thus in and in .

So

=

=