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

Using cylindrical shells, we can think of the sphere as a stack of infinitely thin cylindrical shells. Each shell has a radius equal to the distance from the center of the sphere and a height equal to the circumference of the sphere at that radius. The volume of each shell is then $dV = 2\pi rh\,dr$, where $r$ is the radius and $h$ is the height of the shell, and $dr$ represents an infinitesimal change in radius.

If we integrate over all possible radii from 0 to $r$, we get:

\begin{align*}
V &= \int_{0}^{r} 2\pi rh\,dr \
&= \int_{0}^{r} 2\pi r\cdot 2\pi r\,dr &&	ext{(since }h=2\pi r	ext{)}\
&= \frac{4}{3}\pi r^3
\end{align*}

Thus, the correct answer is B: $V = \frac { 4 } { 3 } \pi r ^ { 3 }$.

Use Cylindrical Shells to Find the Volume of the Solid rrr

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