6.3 Volumes by Cylindrical Shells

In 6.2 which Claire so wonderfully covered, we learned how to find volumes using cross-sections of solids perpendicular to the axis of rotation. We did this using the disk and washer methods:

Disk: Washer:

However, the volumes of certain solids cannot easily be found using these two methods. Sometimes it is necessary to use another method: the method of cylindrical shells. Such can be used when the the perpendicular R and r of a washer depend on the same curve. Instead of using cross-sections perpendicular to the axis of rotation, we can use cross-sections parallel to the axis of rotation. Rotating the parallel cross sections about an axis create shells....

If this curve is rotated about the y-axis, one shell would look like this:

In this next picture of a shell, you can see that the thickness of the shell is the difference between radii.

So, the formula for the volume of a cylindrical shell is:

, where r is the average of the radii and is the difference of the radii.
In words, Volume = (circumference)(height)(thickness). This is easier to understand if you imagine the shell cut and rolled out to form a rectangular solid with length , height h, and width .

Based of this formula for the volume of one shell, we can come up with a formula for the volume of an entire solid made up of many shells. If f is a continuous function on the closed interval [a, b], then the volume of the solid obtained by rotating the graph of f from x = a to x = b about the y-axis is
(where ).

Again, is the circumference, f(x) is the height, and dx is the thickness of the shells.

Now for a sample problem:

Find the volume of the solid obtained by rotating the region bounded by and y= 0

about the line x = 3.

Solution:

First, visualize or sketch the graph. Either with your mind or with your calculator, you can see that the region bounded by the restrictions goes from x=0 to x=1 and lies between a smooth curve and the x-axis (similar to the example on page 458).

Then, draw or picture a vertical cross-section of the region. Rotated around x= 3, the shell has a radius of 3-x, a circumference of , and a height of y or .

So, the volume would be:

Plug in 1 and we get .

Some websites:

http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_02_finding_vol_by_using_cylind_shells.htm

http://pear.math.pitt.edu/Calculus2/week6/6_2li1.html

And another awesome photooo and some awesome pollution:

It looks like RYAN is up next.