A solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis) that lies on the same plane . Here, we will study how to compute volumes of these objects. Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness

## A Volume of Revolution

A solid formed by rotating a curve around an axis.

## Disc Method

The disc method is used when the slice that was drawn is *perpendicular to* the axis of revolution; i.e. when integrating *parallel to* the axis of revolution. The volume of the solid formed by rotating the area between the curves of

If

## Disc Integration

Disc integration about the

The method can be visualized by considering a thin horizontal rectangle at

where

An infinite sum of the discs between

## Shell Method

The shell method is used when the slice that was drawn is *parallel to* the axis of revolution; i.e. when integrating *perpendicular to* the axis of revolution. The volume of the solid formed by rotating the area between the curves of

If

## Shell Integration

The integration (along the

The method can be visualized by considering a thin vertical rectangle at