# MAT 235 New York University Week 9 Applying Fubinis Theorem Worksheet

Description

1
MAT235Y Assignment 3
Due by: Saturday, July 31, 2021 at 1:59PM EDT via Crowdmark
Number of questions: 4
Total points available: 40
1. (8 points) For s > 0, evaluate
Z
F (s) :=
0
(Hint: First write e−sx sin(x)
=
x
R∞
1

e−sx
sin(x)
dx.
x
se−sxy sin(x) dy and apply Fubini’s theorem.)
2
2. (8 points) Using polar coordinates, compute
(a) (4 points) the area of the region lying inside the curve (x2 + y 2 )2 = 2(x2 − y 2 ) and
outside the curve x2 + y 2 = 1 ;
(b) (4 points) and the area of the region lying inside the curve x2 + y 2 = 1 and outside the
curve (x2 + y 2 )2 = 2(x2 − y 2 ).
3
3. (12 points) In this question, we will compare the masses, mB and mE , and moments of
inertia, IB and IE , of both a solid sphere B = {x2 + y 2 + z 2 ≤ 1} and a solid truncated
sphere (that is, sliced to remove a spherical cap) E = {x2 + y 2 + z 2 ≤ 1} ∩ {x ≤ 12 } of
uniform densities, ρB and ρE , that are precessing about the x-axis.
(a) (5 points) Show that the masses of the unit and truncated spheres are mB =
mE = 9ρ8E π.
4ρB
π
3
and
(b) (5 points) The moment of inertia IV of an object V rotating about the x-axis is
ZZZ
(y 2 + z 2 )ρ(x, y, z) dV.
IV =
V
Using cylindrical coordinates, compute the moments of inertia IB and IE of the unit
sphere and the truncated sphere.
(c) (2 points) If unit sphere B and truncated sphere E have the same mass M , but different
uniform densities, then which object will have the larger moment of inertia? Which
will roll down an inclined plane fastest?
4
4. (12 points) Suppose we have an infinitely large 2-dimensional pool of water and at the origin
(0, 0), a small drop of ink at time t = 0. The ink will slowly mix with water, and the
concentration ρ(x, y, t) of ink (mass of ink per volume of water) at any point in space (x, y)
and time t > 0 can be modelled by the equation
ρ(x, y, t) =
1 − 1 (x2 +y2 )
e 4t
.
4πt
(a) (4 points) Show that the total mass of ink in the pool, at any time t > 0, is equal to 1.
(b) (2 points) Suppose this pool has a current (the water is flowing) with constant velocity
~u = hu1 , u2 i. Rewrite the formula for the concentration ρ(x, y, t) (Hint: The ink drop
would travel to point (u1 t, u2 t) at time t and so your new formula of concentration
should just be shift of the original formula.)
(c) (4 points) Consider you are filming with a circular camera of radius R to measure the
total mass of ink. To do so, you must follow the path of the ink droplet. Give a formula
for the total mass inside this moving domain for any time t. (Hint: At a fixed time t,
consider where the domain should be centred to follow the ink in the current.)
(d) (2 points) Calculate the rate of change of the average concentration.
2
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