Due date: Sunday 2/22, 11:59pm
Moments and CoM
01
03
Link to originalFlatCoMMan
Find the center of mass of FlatCoMMan. Assume a constant mass density
. Use additivity of moments.
Solution
Solutions - 0110-03
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:
Link to original
02
04
Link to originalCoM from Simpson’s
Use Simpson’s rule (with 6 subintervals) to estimate the CoM of this region:
You will need to estimate
and and with three separate integrals. You can use a calculator for your arithmetic.
Solution
Solutions - 0110-04
(1) Simpson’s Rule formula:
(2) Simpson’s for total mass
: Therefore:
So:
(3) Simpson’s for moment to
-axis: Integral formula:
Approximate with
:
(4) Simpson’s for moment to
-axis: Integral formula:
Approximate with
:
(5) Compute CoM:
Link to original
Improper integrals
03
02
Link to originalProper vs. improper
For each integral below, determine whether it is proper or improper, and if improper, explain why.
(a)
(b) (c) (d)
(e) (f)
Solution
Solutions - 0130-02
(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
Link to originalas . (Same indeterminate form as (e).)
04
03
Link to originalGabriel’s Horn - Volume and surface of revolution
The curve
for is rotated about the -axis. The resulting shape is Gabriel’s Horn. (a) Find the volume enclosed by the horn by evaluating a convergent improper integral.
(b) Show that the surface area of the horn is infinite by applying comparison to a
-integral which is divergent.
Solution
Solutions - 0130-03
Volume:
Surface area:
But notice this:
But
diverges! So by the comparison test,
Link to originaldiverges as well.
05
01
Link to originalComparison test
Use the comparison test to determine whether the integral converges:
Solution
Solutions - 0130-01
(1) Find comparable integrand:
Higher power dominates for large
: Therefore, compare to
.
(2) Make comparison:
And:
because it is a
-integral with . By the Comparison Test, we conclude that:
Link to original
06
05
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 - 0130-05
(a)
(1) Definition of improper integral:
(2) Antiderivative and limit:
Note A: Use L’Hopital:
(b)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(c)
(1) Definition of improper integral:
(2) Antiderivative and limit:
Link to original