Question

How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=1/x, y=0, x=1, and x=2 about the y-axis?

Answer 1

Please see the explanation section, below.

Explanation:

Here is the region (blue) with a thin slice taken parallel to the axis of rotation. To set up for cylindrical shells. (The slice has black edges.)

The volume of a representative shell is `2pirh*"thickness"`

In this case, we have

radius `r = x` (the dashed black line),

height `h = 1/x` and

`"thickness" = dx`.

`x` varies from `1` to `2`, so the volume of the solid is:

`int_1^2 2pi x(1/x)dx =2 piint_1^2 dx = 2pi[x]_1^2 = 2pi`