# Volumes of Solids of Revolution

## Rotation about Horizontal Axis

### Volumes by Disks

This is the simplest case. Since $dV=(area of face) (thickness)$, we have that $V=∫dV$ $V=∫_{a}π[f(x)]_{2}dx$

### Volumes by Washer

This is an extension of the disc method, where $V=∫dV$, but now we are dealing with a hollowed object. $dV=(πr_{outer}−πr_{inner})dx$

In the case the line is down by $k$, we have that $dV=π((f(x)+k)_{2}−k_{2})dx$

We have that

## Rotation about Vertical Axis

### Volumes by Shells

In some cases, volumes of rotation are not conveniently described by the disk method. In these cases, we use the shell method.

$dV=πr_{outer}h−πr_{inner}h$ $dV=π[2xdx+(dx)_{2}]f(x)$ Since $(dx)_{2}$ is infinitesimally small, we can ignore that value. $V=∫dV$ $V=∫_{a}2πxf(x)dx$