Examples

Revolution of a triangle

A rotation-symmetric 3D body has cross section given by the region between , , , and is rotated around the -axis. Find the volume of this 3D body.

Solution

(1) Cross-section region:

Bounded above-right by . Bounded below-right by . These intersect at .

Bounded left by .

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(2) Set up integral:

Rotated around -axis, therefore use for integration variable (shells!). Formula:

Domain is .

because shell radius is the -distance from to the shell position.

Height:

is limit of which equals here, so .

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(3) Evaluate integral:

Revolution of a sinusoid

Consider the region given by revolving the first hump of about the -axis. Set up an integral that gives the volume of this region using the method of shells.

Solution

(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 .

Use IBP formula:

Compute first term:

Compute integral term:

So the answer is .