2 Volumes generated by rotating curves about the y-axis

We can obtain a different solid of revolution by rotating a curve around the $y$ -axis instead of around the $x$ -axis. See Figure 10.

Figure 10 :

To find the volume of this solid it is divided into a number of circular discs as before, but this time the discs are horizontal. The radius of a typical disc is $x$ and its thickness is $\delta y$ . The volume of the disc will be $\pi x^2\delta y$ .

The total volume is found by summing these individual volumes and taking the limit as $\delta y\to 0$ . If the lower and upper limits on $y$ are $c$ and $d$ , we obtain for the volume:

$\underset{\delta y\to 0}{lim}{\sum }_{y=c}^{y=d}\pi x^2\delta y$ which is the definite integral ${\int \; <!--nolimits\; -->}_c^d\pi x^{2}dy$

Task!

Find the volume generated when the graph of $y=x^2$ between $x=0$ and $x=1$ is rotated around the $y$ -axis.

Using Key Point 6 write down the required integral:

Answer

This integral can be written entirely in terms of $y$ , using the fact that $y=x^2$ to eliminate $x$ . Do this now, and then evaluate the integral:

Answer

Exercises

1. The curve $y=x^2$ for $1<x<2$ is rotated about the $y$ -axis. Find the volume of the solid formed.
2. The line $y=2-2x$ for $0\le x\le 2$ is rotated around the $y$ -axis. Find the volume of revolution.

Answer