1.8.3 Volume of Revolution
In this topic we will learn how to:
- use definite integration to find a volume of revolution about one of the axes
We can use integration to find the volume produced by a graph when it rotates about an axis . This is known as the volume of revolution. To find this we use the two formulae below,
It is derived from the formula of volume of a cylinder, , hence the similarities. The first equation is used when we’re rotating about the -axis. The second equation is used when we’re rotating about the -axis.
Let’s apply this to some past paper questions.
1. The diagram shows part of the curve with equation . The shaded region enclosed by the curve, the -axis and the line is rotated through about the -axis. Find the volume obtained. (9709/12/F/M/20 number 3)
The volume of revolution of this graph would look like this,
Since we’re rotating about the -axis we will use the formula,
This means we need to find in terms of since we are working with respect to ,
Let’s substitute in ,
We are already given the limits in terms of , so let’s substitute them in,
Note: If the limits are not in terms of and you’re rotating about the -axis, use the equation of the curve, to convert them to be in terms of .
Now let’s integrate,
is a constant so you can move it outside the integral sign to make the integration easier,
Substitute in the limits,
Therefore, the final answer is,
2. The diagram shows part of the curve with equation and the lines and . The shaded region is enclosed by the curve and the lines is rotated through about the -axis. Find the volume obtained. (9709/12/M/J/21 number 9)
The volume of revolution would look something like this,
Notice that we only want the shaded region which is bounded by the curve and the line . For that reason, we also need to consider the cylinder formed by the line when it rotates,
Let’s first find the volume produced by the curve,
Substitute in ,
From the diagram, we can tell that one of our limits is . The other limit is at , so we need to convert it to be in terms of ,
Therefore, our limits are and ,
Now let’s integrate,
Substitute in the limits,
Now let’s find the volume produced by the cylinder due to the line ,
Therefore,
Therefore, the final answer is,