Exercise solutions - Unit 01

Volume using cylindrical shells

02 - Revolution of a sinusoid

(1) 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:

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(2) 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 .

Integration by parts

04 - Hidden A

(1) Choose .

Because Log is farthest right in LIATE.

It follows that we must choose .

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(2) Compute and .

We have and .

Obtain chart:

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(3) Plug into IBP formula.

Plug in all data:

Integrate:

Final answer is: