Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the line x = 2.
Answer
When we rotate R around the line x = 2 we get the solid
The shell at position x has height . Its radius is the distance from the curve to the axis of rotation x = 2.
The radius is
2 – x.
The volume of the shell is
The shells go from x = 0 to x = 1. This means we need to integrate from 0 to 1 to get the volume of the solid, or
Example 2
Once again, we'll let R be our favorite region.
Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the line y = 1.
Answer
When we rotate R around the line y = 1 we get the solid
The shell at height y looks like this. The radius of the shell is the distance from the edge of the cylinder (at y) to the axis of rotation (y = 1). So
radius = 1 – y.
The height of the shell is the distance from the curve to the line x = 1, or
height = 1 – x = 1 – y2.
The thickness of the shell is Δ y, so the volume of the shell is
2π(1 – y)(1 – y2) Δ y.
Since the shells go from y = 0 at the outside of the solid to y = 1 at the inside of the solid, the volume of the solid is
That's enough of that region. Let's do some problems with a different one, just to mix things up.
Example 3
Let R be the region in the first quadrant bounded by the graph y = x2, the y-axis, and the line y = 4. Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the y-axis.
Answer
The region R looks like
When we rotate the region R around the y-axis we get the solid
Since the y-axis is the axis of rotation, the shells will have their opening around the y-axis. The shell at position x looks like
The radius of the shell is the distance from the edge of the shell to the y-axis, so
radius = x.
The height of the shell is the distance from the curve y = x2 to the line y = 4, or
height = 4 – x2.
The volume of a shell is
2π(x)(4 – x2) Δ x.
The variable x goes from 0 at the center of the solid to 2 at the outside of the solid, so the volume of the solid is
Example 4
Let R be the region in the first quadrant bounded by the graph y = x2, the y-axis, and the line y = 4. Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the line x = -1.
Answer
The region R looks like
When the region R is rotated around the line x = -1, we get a solid that looks like
The shells run parallel to the axis of rotation, so a shell at position x will look like
The radius of the shell is the distance from the edge of the shell to the axis of rotation x = -1, so
radius = x + 1.
The height of the shell is the distance from the curve y = x2 to the line y = 4, which is
height = 4 – x2
(the height is the same as in the previous problem). The volume of a shell is
2π(x + 1)(4 – x2) Δ x.
Shells go from x = 0 at the inside edge of the solid to x = 2 at the outside edge of the solid, so the volume of the solid is
Example 5
Let R be the region in the first quadrant bounded by the graph y = x2, the y-axis, and the line y = 4. Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the x-axis.
Answer
The region R looks like
When the region R is rotated around the x-axis we get
The shell at position y is centered around the axis of rotation:
The radius of the shell is y, and the height of the shell is . The volume of the shell is
The shells go from y = 0 at the inside of the solid to y = 4 at the outside of the solid, so the volume of the solid is
Example 6
Let R be the region in the first quadrant bounded by the graph y = x2, the y-axis, and the line y = 4. Use the shell method to write an integral expression for the volume of the solid obtained by rotating R around the line y = 5.
Answer
The region R looks like
When the region R is rotated around the line y = 5, we get the solid
The shell at position y looks like
The radius of the shell is the distance from the edge of the shell to the axis of rotation y = 5, so
radius = 5 – y.
The height of the shell is x, which is the same as . So the volume of the shell is
The shells still go from y = 0 to y = 4, so the volume of the solid is
. Its radius is the distance from the curve 
to the axis of rotation x = 2.
to the line x = 1, or
. The volume of the shell is
. So the volume of the shell is
