Disks and washers about ordinary and shifted axes

Understand the idea

For rotation, radius is distance from the rotation axis. A disk has one radius; a washer removes an inner disk from an outer disk. Both radii must be measured in the same slicing direction. A shifted axis changes the radius expressions, not the general formula. Choose vertical slices for rotation about a horizontal axis and horizontal slices for rotation about a vertical axis when using disks or washers.

V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx

Worked example

Rotate the region between $\displaystyle y=x$ and $\displaystyle y=x^2$ , $\displaystyle 0\le x\le1$ , about the $\displaystyle x$ -axis.

The outer radius is

\displaystyle x

and inner radius is

\displaystyle x^2

. The washer area is

\displaystyle \pi(x^2-x^4)

, so volume is

\displaystyle \pi(1/3-1/5)=2\pi/15

.

Try it yourself

Rotate the region below $\displaystyle y=x$ and above $\displaystyle y=0$ for $\displaystyle 0\le x\le1$ about $\displaystyle y=2$ . Set up the volume.

Show solution and reasoning

The outer radius is $\displaystyle 2$ and inner radius is $\displaystyle 2-x$ . The volume is $\displaystyle \pi\int_0^1[4-(2-x)^2]\,dx=5\pi/3$ .