Due date: Thursday 2/19, 11:59pm
Moments and CoM
01
01
Link to originalCoM of a house
A “house” is the region bounded by the (non-regular) pentagon with vertex points at
, , , , . Find the CoM of the house using additivity of moments.
Solution
Solutions - 0110-01
(1) Compute masses:
(2) Consider symmetries of rectangle:
Therefore
and: Therefore:
(3) Consider symmetry of triangle:
(4) Compute
by integration:
(5) Optional step: infer
:
(6) Additivity of moments:
(7) Compute CoM:
Thus:
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02
02
Link to originalCoM of region between curves
Find the CoM of the region between the graph of
and the graph of over .
Solution
Solutions - 0110-02
Option 1: In terms of
. (1) Mass of region:
(2) Integral formula for
:
(3) Integral formula for
:
(4) Compute
:
Option 2: In terms of
. (1) Mass:
(2)
:
(3)
: Link to original
Improper integrals
03
04
Link to originalComputing improper integrals
For each integral below, give the limit interpretation of improper integral and then compute the limit. Based on that result, state whether the integral converges. If it converges, what is its value?
(a)
(b) (c)
Solution
Solutions - 0130-04
(a)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(b)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(c)
(1) Definition of improper integral:
(2) Antiderivative and limit:
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