W06 - HW Solutions Reg

04

Assume for all of these.

The value of does not affect the CoM point if is a constant.

Region 1:

Region 2:

Region 3:

Region 4:

Region 5:

Region 6:

Region 7:

FlatCoMMan:

05

(1) Simpson’s Rule formula:

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(2) Simpson’s for total mass :

Therefore:

So:

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(3) Simpson’s for moment to -axis:

Integral formula:

Approximate with :

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(4) Simpson’s for moment to -axis:

Integral formula:

Approximate with :

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(5) Compute CoM:

06

(a) Improper: integrand as .

Note: this converges too, since it’s a -integral to zero with .

(b) Proper: no source of infinity.

Note: automatically converges.

(c) Improper: as .

Note: this diverges. Antiderivative is as .

(d) Improper: infinite upper bound.

Note: this diverges. Antiderivative is which has no limit as .

(e) Improper: infinite upper bound.

Note: this diverges. Antiderivative is as . (L’Hopital’s rule, indeterminate form.)

(f) Improper: infinite integrand at .

Note: this converges. Antiderivative is as . (Same indeterminate form as (e).)

07

Volume:

Surface area:

But notice this:

But diverges!

So by the comparison test, diverges as well.

09

(a)

(1) Definition of improper integral:

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(2) Antiderivative and limit:

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(b)

(1) Definition of improper integral:

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(2) Antiderivative and limit:

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(c)

(1) Definition of improper integral:

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(2) Antiderivative and limit:

10

(a)

(1) Definition of improper integral:

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(2) Antiderivative and limit:

Note A: Use L’Hopital:

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(b)

(1) Definition of improper integral:

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(2) Antiderivative and limit:

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(c)

(1) Definition of improper integral:

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(2) Antiderivative and limit: