Volume of revolution formula

Rotation about the x axis

If we have a curve with the equation , we can calculate the volume of the (full) rotation about the x axis using the formula:

It usually helps to find first before trying to integrate and find the volume.

Rotation about the y axis

If instead we want to rotate the curve of about the y axis by 360 degrees (or radians), then we just swap the and :

This does mean you need to rearrange the function of to make the subject, then square it.

Rotation of less than

If we’re not rotating the full way, then we may want to find a fraction of the full volume. This depends entirely on what the shape looks like. For example, if we already have a full ‘semicircle’ shape in the top two quadrants on the graph, then we’ll have fully created all the volume after just half a revolution about the y-axis (because it’s ‘two sided’ and symmetrical).

The easiest thing to do is think logically about how much it needs to be rotated until it’s created its full volume.

If it needs to rotate 360 degrees to create its full volume, and we’re only rotating it 180 degrees, then we can just take half of the full volume: