First, we note that the integrand f ( x ) = r 2 − x 2 {\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}} is an even function, because
f ( − x ) = r 2 − ( − x ) 2 = r 2 − x 2 = f ( x ) {\displaystyle f(-x)={\sqrt {r^{2}-(-x)^{2}}}={\sqrt {r^{2}-x^{2}}}=f(x)} ,
therefore, by Hint (1) we have
2 ∫ − r r r 2 − x 2 d x = 2 ( 2 ∫ 0 r r 2 − x 2 d x ) = 4 ∫ 0 r r 2 − x 2 d x {\displaystyle 2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx=2\left(2\int _{0}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\right)=4\int _{0}^{r}{\sqrt {r^{2}-x^{2}}}\,dx}
is an even function, because
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