Problem: Consider the region R under the the graph of the function
Find the volume of the solid obtained by revolving this region about the y-axis.
Solution: We start with a picture:
Since the region never stops at the right end, the resulting solid will be also unbounded. We may imagine that it looks like a tent supported in the middle and spreading to infinity symmetrically in all directions, getting closer and closer to the ground. Such an unbounded solid may have an infinite volume, we will see it after we check on the convergence of the relevant integral. Since the situation is straightforward, we may start calculating right away using the shell method (we have a horizontal axis of rotation):
The integration by parts was standard, but would we save some work by switching the axes?
Now we should use the disc method:
This integral is also solved using integration by parts, so it would take about as much effort as the previous solution.