Use cylindrical shells to find the volume of the solid. A sphere of radius $r$ .
A) $V=\frac{1}{3} \pi r^{3}$
B) $V=\frac{4}{3} \pi r^{3}$
C) $V=\frac{7}{3} \pi r^{3}$
D) $V=\frac{2}{3} \pi r^{3}$
E) $V=\frac{5}{3} \pi r^{3}$

Question

Quizplus · Accepted Answer

We can use cylindrical shells to find the volume of the sphere. For each shell, the radius will be equal to the distance from the axis of rotation (which is also the center of the sphere) to the shell. The height of each shell will be equal to the circumference of the sphere at that radius, since it sweeps out a cylinder as it revolves around the axis. Therefore, the volume of each shell will be given by:

$$
dV = 2 \pi r \cdot 4 \pi r^{2} dr
$$

Integrating from 0 to r, we get:

$$
\begin{aligned}
V &= \int_{0}^{r} dV \
&= \int_{0}^{r} 2 \pi r \cdot 4 \pi r^{2} dr \
&= 2 \pi \cdot 4 \pi \int_{0}^{r} r^{3} dr \
&= 2 \pi \cdot 4 \pi \cdot \left[\frac{1}{4} r^{4}ight]_{0}^{r} \
&= \frac{4}{3} \pi r^{3}
\end{aligned}
$$

Therefore, the volume of the sphere is given by $\frac{4}{3} \pi r^{3}$, which corresponds to choice (B).

Use Cylindrical Shells to Find the Volume of the Solid rrr

Use Cylindrical Shells to Find the Volume of the Solid $r$