Let R be the region bounded by #y = 1/x#, #y = x^2#, #x = 0#, and #y = 2# and revolved about the x axis. How do you find the volume of rotation using: a) the method of cylindrical shells; b) the method of circular disks?
Because we are looking at revolution about an horizontal axis, using cross-section (See videos 1 through 6 here if you need a refresher) is more natural, so let us start with that.
Here is the region we are rotating about the
The first thing to understand is where the curves intersect.
So the expression for the area
On the other hand if
As a result, the volume of the resulting solid of revolution is
On other hand,
Hence, altogether, we obtain
Considering the length of the answer, I am leaving the cylindrical shell part to another time.