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The problem: The region between y=1/x and the lines x=1/H, y=H, and y=1/H is rotated around the x-axis. Show, using disks/washers that the volume of this solid is
V = pi(H^2-1)^2/H^3
First, started this problem by experimenting with different values of H. I let H=2 and H=3 just to see any patterns. I noticed that the graphs of y=H and y=1/x always intersect at the point x=1/H. I also noticed the graphs of y=1/x intersects y=1/H at H. So I think my limits of integration would be from 1/H to H.
Second, I used washers. It seemed like the solid would look a funnel and I let the outer radius = 1/x and the inner radius = 1/H
Thirdly, adding it all up together I wanted to find pi multiplied by the integral of ((1/x)^2 - (1/H)^2)dx from 1/H to H.
I wanted to fix this up a bit and find common demoninators so I found the integrand to be (H^2-x^2)/(x^2*H^2)
This is where I am stuck at. I think I am on the right track because I used my calculator to find the answer and I think I came up with the right answer. I just don't know how to integrate that function. Maybe I have to put 1/H in terms of x?
