Question

Prove that ${\int }_{-a}^a{x}^3\sqrt{a^2-x^2}dx=0$

Answer 1

${\int }_{-1}^a{x}^3\sqrt{a^2-x^2}dx$

$f\left(x\right)=x^3\sqrt{a^2-x^2}$

$f(-x)=(-x{)}^3\sqrt{a^2-(-x{)}^2}$

$=-x^3\sqrt{a^2-x^2}$

$=-f\left(x\right)$

$\therefore f\left(x\right)$ is odd function.

$\therefore {\int }_{-a}^a{x}^3\sqrt{a^2-x^2}dx=0$.