Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 297: 20

$V$ = $\frac{2}{3}r^{3}$

base = $2\sqrt {r^{2}-x^{2}}$ height = $x$ $V$ = $\int_0^{r}2x\sqrt {r^{2}-x^{2}}dx$ $u$ = ${r^{2}-x^{2}}$ $du$ = $-2xdx$ $V$ = $\int_0^{r^{2}}\sqrt {u}du$ $V$ = $\frac{2}{3}u^{\frac{3}{2}}|_0^{r^{2}}$ $V$ = $\frac{2}{3}r^{3}$