Answer
$$
f(x)=\sqrt {r^{2}-x^{2}}
$$
The function has as its graph a semicircle of radius $r$ with center at$(0,0)$.
The volume that results when the semicircle is rotated about the $x$-axis is given by :
$$
\begin{aligned} V &=\pi \int_{-r}^{r}(f(x))^{2} d x\\
&=\pi \int_{-r}^{r}(\sqrt{r^{2}-x^{2}})^{2} d x \\
&= \frac{4 \pi r^{3}}{3}.
\end{aligned}
$$
Work Step by Step
$$
f(x)=\sqrt {r^{2}-x^{2}}
$$
The function has as its graph a semicircle of radius $r$ with center at$(0,0)$.
The volume that results when the semicircle is rotated about the $x$-axis is given by :
$$
\begin{aligned} V &=\pi \int_{-r}^{r}(f(x))^{2} d x\\
&=\pi \int_{-r}^{r}(\sqrt{r^{2}-x^{2}})^{2} d x \\
&=\pi \int_{-r}^{r}\left( r^{2}-x^{2}\right) d x \\
&=\left.\pi\left(r^{2}x-\frac{x^{3}}{3}\right)\right|_{-r} ^{r} \\ &=\pi\left(r^{3}-\frac{r^{3}}{3}\right)-\pi\left(-r^{3}+\frac{r^{3}}{3}\right) \\
&= \frac{4 \pi r^{3}}{3}.
\end{aligned}
$$
