Solutions - 0030-03

(a)

(1) Write down shells formula:

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(2) Define the cross section region:

Bounded above by . Bounded below by .

Bounded left by . Bounded right by intersection at line .

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(3) Define and and :

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(4) Plug into shells formula and compute:

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(b)

(1) Write down washers formula using :

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(2) Rewrite bounding equations in terms of :

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(3) Determine region boundary data:

Bounded above by . Bounded below by . Intersection at .

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(4) Determine in two components, with the dividing line:

Note that for both regions. These are disks.

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(5) Compute the integral:

(6) Why are shells preferable?

1. Only need one integral.
2. Don’t need to rewrite boundary equations in terms of .