W06 - HW Solutions Step

01

(1) Compute masses:

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(2) Consider symmetries of rectangle:

Therefore and:

Therefore:

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(3) Consider symmetry of triangle:

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(4) Compute by integration:

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(5) Optional step: infer :

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(6) Additivity of moments:

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(7) Compute CoM:

Thus:

02

Option 1: In terms of .

(1) Mass of region:

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(2) Integral formula for :

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(3) Integral formula for :

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(4) Compute :

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Option 2: In terms of .

(1) Mass:

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(2) :

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(3) :

03

(1) Find comparable integrand:

Higher power dominates for large :

Therefore, compare to .

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(2) Make comparison:

And:

because it is a -integral with .

By the Comparison Test, we conclude that: