Volumes Using Shells

A shell is a hollow tube. To calculate volume using shells, we take shells of graduated radii and fill the solid of revolution with them. We only want the volume of the material in each shell, not what the shell itself might hold. We calculate the volume of each shell and add them all up to get the total volume. This will create a Riemann Sum which we can transfer to an integral.

The formula for the volume of a shell is V = 2 p r h w
w is our dx or dy. The width of the height of the shell tells us which to use.

Problem:

Find the volume of the region bounded by y = e^2x, x = ln 4, and y = e ^-x, as it is rotated about the
line x = -2.

Solution:

Graph the equations and shade in the region indicated.
Draw a representative shell in the shaded area and its rotation around the line x = -2.
Find an expression for the radius of the shell: r = x + 2
Find an expression for the height of the shell: h = e^2x - e ^-x
Decide if this is a dx or dy case. (The differential in the height is an x-value, so dx)
Put these pieces into the formula and calculate the integral over the shaded area.

ln 4

2p ò (x + 2)(e^2x - e ^-x) dx = 20.437

(You may use your calculator to calculate the

integral value.)