I don't think that you have enough bounds on your region...
The Problem
The finite region bounded by the curve , the y-axis, and the line , is rotated through 360 degrees about the x-axis. Show that the volume of the solid formed is .
Attempt
Volume = , with limits
This becomes: Volume = , with limits
(We can take pi outside the integral and multiply the result by it later).
Integrating term by term:
.
Substituting :
Substituting :
Subtracting:
.
So my answer is:
.
So I don't know how they've reached the answer given.
Any hints?
I'm not exactly sure what you mean by this. This is from a textbook, so do you mean that the information given does not give enough bounds to determine the volume of the region, or do you mean that the bounds of 0 and ln 2 that I have used are wrong?
Also, I found that if you multiply out the answer given, you get . My answer is equivalent to , so I just can't see how they got the second part.
Anyone else have any ideas?