Chapter 5.9, Problem 69ES

(a) Prove that if $f$ is an odd function, then

${\displaystyle {\int }_{-a}^{a}f\left(x\right)dx}=0$

(b) Prove that if $f$ is an even function, then

${\displaystyle {\int }_{-a}^{a}f\left(x\right)dx}=2{\displaystyle {\int }_0^{a}f\left(x\right)dx}$

and give a geometric explanation of this result.