Exercise solutions

Volume using cylindrical shells

Set up the integral for shells.

Integration variable: , the distance of a shell to the -axis.

Then and , the height of a shell.

Bounds: one hump is given by .

Thus:

Perform the integral using IBP.

Choose and since is A and is T.

Then and .

Recall IBP formula:

Insert data in IBP formula:

Compute first term:

Compute integral term:

So the answer is .

Choose .

Because Log is farthest right in LIATE.

It follows that we must choose .

Compute and .

We have and .

Obtain chart:

Plug into IBP formula.

Plug in all data:

Integrate:

Final answer is: