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

To use cylindrical shells, we imagine slicing the sphere into thin, vertical strips and then integrating their volumes. Each shell has the same height as the sphere and a circumference equal to the circumference of the sphere.

The volume of a cylindrical shell is given by $2\pi rh\Delta{r}$. In this case, $h$ is the height of the sphere, which is $2r$. The radius of each cylindrical shell is $r - \Delta{r}/2$.

So, the integral for the volume becomes:

$$
\begin{aligned} V &= \int_0^r 2\pi(r-\frac{\Delta{r}}{2})\cdot 2r \cdot \Delta{r} \ &= 4\pi \int_0^r (r-\frac{\Delta{r}}{2})\cdot \Delta{r} \ &= 4\pi \int_0^r r\Delta{r} - 2\pi\int_0^r \frac{1}{2}\Delta{r}^2 \ &= 4\pi [\frac{1}{2}r^2] - 2\pi[\frac{1}{2}\cdot(\frac{1}{3}r^3)] \ &= \frac{4}{3}\pi r^3 \end{aligned}
$$

Therefore, the correct choice is B.

Use Cylindrical Shells to Find the Volume of the Solid rrr

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