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W01 - Examples

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01 - 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 Define the cross section region.

Bounded above-right by .

Bounded below-right by .

These intersect at .

Bounded at left by .

Define range of integration variable.

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

Integral over :

Interpret .

Radius of shell-cylinder equals distance along :

Interpret .

Height of shell-cylinder equals distance from lower to upper bounding lines:

Interpret .

is limit of which equals here so .

Plug data in volume formula.

Insert data and compute integral:

02 - 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

03 - A and T factors

Compute the integral:

Solution Choose .

Set because simplifies when differentiated. (By the trick: is Algebraic, i.e. more “”, and is Trig, more “”.)

Remaining factor must be :

Compute and .

Derive :

Antiderive :

Obtain chart:

Plug into IBP formula.

Plug in all data:

Compute integral on RHS:

Note: the point of IBP is that this integral is easier than the first one!

Final answer is:

04 - Hidden A

Compute the integral:

Solution