Volume of Revolution Method Two (Shell Method)

I am going to use the same example as I did for Method One (Disc or Washer Method).

If we rotate the shaded region about the $x-$ axis, we get an open hollow cylinder (like a pipe).

The width of the integral is $\delta y$ and the midpoint is $y$ .

The height of the cylinder is $x$ , but we need it in terms of $y$ , hence $x=f(y)$

The volume of the hollow cylinder is the volume of the outer cylinder subtract the volume of the inner cylinder.

$\begin{equation*}V=\pi (y+\frac{\delta y}{2})^2f(y)-\pi (y-\frac{\delta y}{2})^2 f(y)\end{equation}$

$\begin{equation*}V=\pi f(y)((y+\frac{\delta y}{2})^2-(y-\frac{\delta y}{2})^2)\end{equation}$

Which we can expand using a difference of squares.

$\begin{equation*}V=\pi f(y)(y+\frac{\delta y}{2}+y-\frac{\delta y}{2})(y+\frac{\delta y}{2}-y+\frac{\delta y}{2})\end{equation}$

$\begin{equation*}V=\pi f(y)(2y \delta y)\end{equation}$

$\begin{equation*}V=2\pi yf(y)\delta y\end{equation}$

The volume of the entire sold will be

$\begin{equation*}V=\Sigma_{y=a}^b 2 \pi yf(y)\delta y\end{equation}$

As $\delta y \rightarrow 0$

$\begin{equation*}V=\lim\limits_{\delta y \to 0}\Sigma_{y=a}^b 2 \pi yf(y)\delta y=\int_a^b 2\pi yf(y) dy\end{equation}$

Even though we are rotating the line about the $x-$ axis, we are integrating with respect to the $y-$ axis.

Example

Find the volume of the solid generated by revolving the region between $y=x^2$ and $y=2x$ about the $y-$ axis.

If we are rotating about the $y-$ axis, we will integrate with respect to $x$ .

$\begin{equation*}V=2\pi \int x f(x) dx\end{equation}$

Hence

$\begin{equation*}V=2\pi\int_0^2 x(2x-x^2) dx\end{equation}$

$\begin{equation*}V=2\pi \int_0^2 (2x^2-x^3) dx\end{equation}$

$\begin{equation*}V=2\pi (\frac{2x^3}{3}-\frac{x^4}{4}]_0^2\end{equation}$

$\begin{equation*}V=2\pi (\frac{2}{3}\times 8-\frac{1}{4}\times 16 )\end{equation}$

$\begin{equation*}V=32 \pi(\frac{1}{3}-\frac{1}{4})\end{equation}$

$\begin{equation*}V=\frac{8\pi}{3}\end{equation}$

Let’s check with method one.

$x^2=y$ and $x=\frac{y}{2}$

$\begin{equation*}V=\pi \int_0^4 y-\frac{y^2}{4} dy\end{equation}$

$\begin{equation*}V=\pi (\frac{y^2}{2}-\frac{y^3}{12})]_0^4\end{equation}$

$\begin{equation*}V=\pi(8-\frac{16}{3})\end{equation}$

$\begin{equation*}V=\frac{8\pi}{3}\end{equation}$

I try to pick the method that makes the integration easier.