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) Define the cross section region.

Bounded above-right by .

Bounded below-right by .

These intersect at .

Bounded at left by .

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(2) Define range of integration variable.

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

Integral over :

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(3) Interpret .

Radius of shell-cylinder equals distance along :

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(4) Interpret .

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

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(5) Interpret .

is limit of which equals here so .

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(6) Plug data in volume formula.

Insert data and compute integral: