Area, Volume, and Arc Length
No, we aren't talking about Sally's seashells by the seashore. We are talking about the another method for finding the volumes of weird 3-D things by chopping them up into funny-shaped pieces. It is called the cylinder method or shell method. Instead of chopping the solid into disks or washers perpendicular to the axis of rotation, we carve it into cylindrical shells that surround the axis of rotation.
Let's get some visual help. We can take the region surrounded by , the line x = 1, and the x-axis:
If we rotate that region around the y-axis, and then remove a cylinder-shaped shell from around the axis of rotation, we get
This shell is like a piece of paper that's rolled together so its edges just barely touch. It's called a shell because there's nothing inside it. Like a toilet paper tube, there's a little bit of stuff around the edge and a lot of empty space in the middle. Ack! We ran out of toilet paper.
We can find the volume of the solid by adding up the volumes of the infinitely many shells it's made of. This is similar to when we found the area of circle by chopping it into rings.
We have to answer a simple question first: what is the volume of a shell? If a shell is like a toilet paper tube, then we can unroll the tube. The height of the shell is the height of the unrolled cardboard. The circumference of the shell is the width of the unrolled paper. This means the area of one side of the paper is
The tube is very thin, with a thickness Δ r, where r is the variable describing the position of the shell. This means the volume of the piece of tubular shell, is
(2πrh) Δ r.
To find the volume of the solid, we integrate over whatever values are reasonable for the variable of integration.
Let R be the region surrounded by , the line x = 1, and the x-axis. Use the shell method to find the volume of the solid obtained by rotating R around the y-axis.
We already know what the region R and the solid look like:
We can use x as the variable of integration. The shell at position x has radius x and height .
The volume of this shell is
We have shells from x = 0 to x = 1, the limits of integration. The volume of the solid is
When the axis of rotation isn't one of the coordinate axes, we have to do a little more work to figure out the radius of the shells. As with the disks and washers, we want to draw pictures to do this.