Question

$\underset{-\pi }{\overset{\pi }{\int }}|\pi -|x\left|\right|dx$ is equal to

• A. ${\pi }^2$
• B. $\frac{{\pi }^2}{2}$
• C. $\sqrt{2}{\pi }^2$
• D. $2{\pi }^2$

Answer 1

The correct option is A ${\pi }^2$

$I=\underset{-\pi }{\overset{\pi }{\int }}|\pi -|x\left|\right|dx$
Here, $|\pi -|x\left|\right|$ is an even function.

$\therefore I=2\underset{0}{\overset{\pi }{\int }}|\pi -|x\left|\right|dx$
$=2\underset{0}{\overset{\pi }{\int }}(\pi -x)dx$
$=2{[\pi x-\frac{x^2}{2}]}_0^{\pi }$
$=2\left[\frac{{\pi }^2}{2}\right]={\pi }^2$