Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

Quizplus · Accepted Answer

The region is bounded by the curves $y=x^2$ and $y=4$ and is rotated about the $y$-axis.

First, we need to find the points of intersection between the two curves:

$$x^2=4\quad\Rightarrow\quad x=\pm 2.$$

Thus, the region we are rotating is as follows:

[asy]
size(150);
import graph;
real f(real x)
{
	return x^2;
}
fill((2,0)--(2,4)--(-2,4)--(-2,0)--cycle,lightcyan);
draw(graph(f,-2.2,2.2),linewidth(1bp));
draw((-2.2,0)--(2.2,0),Arrows(4));
draw((0,-0.2)--(0,4.2),Arrows(4));
label("$y=x^2$",(2.2,f(2)),E);
label("$y=4$",(2.2,4),E);
[/asy]

When we rotate this region about the $y$-axis, we get a solid that is a "shell" of thickness $dx$. The volume of each shell is the circumference of the shell times the height of the shell times the thickness:

$$dV=2\pi x\cdot(4-x^2)\cdot dx.$$Thus, the total volume is given by the integral:

$$\int_{-2}^2 2\pi x\cdot(4-x^2)\cdot dx.$$Evaluating this integral gives:

\begin{align*}
\int_{-2}^2 2\pi x\cdot(4-x^2)\cdot dx&=\left.\pi\left(4x^2-\frac{x^4}{4}ight)ight|_{-2}^2\
&=\pi(16-4-\left(-16+\frac{4}{4}ight))\
&=\boxed{24\pi}.
\end{align*}

Find the Volume of the Solid Obtained by Rotating the Region