Rotate about the x-axis with respect to dy

Question

How would I rotate the region bounded by $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ along the x-axis in terms of dy. I have already solved this problem in terms of dx see here

Here is the graph:https://www.desmos.com/calculator/rij8mhgszd

Answer 1

Integrating with respect to $y$ is (in this case) a little more complicated. We use the Method of Cylindrical Shells.

The solid consists of two parts: (i) the part obtained by rotating the part of the region that lies below $y=4$ and (ii) the part obtained by rotating the part of the region that lies above $y=4$.

For (i) we could integrate but there is no need to do so. We get a cylinder with radius $4$ and height $1$, and therefore volume $16\pi$.

For (ii), take a thin slice of width "$dy$" at height $y$, where $y$ is between $4$ and $5$. Rotate the slice about the $x$-axis. We get a cylindrical shell of "thickness" $dy$, radius $y$, and "height" $1-x$, which is $1-\sqrt{y-4}$. Thus the volume of (ii) is $$\int_{y=4}^5 2\pi y\left(1-\sqrt{y-4}\right)\,dy.$$

Calculate, and add the $16\pi$ obtained earlier.