Homework 12
MATH 2500 - Swinarski
Due at 6PM, Wednesday, April 8
(1) Find the center of mass of a thin plate which is bounded by the curves x=y2and
x= 2y−y2if the density at the point (x, y) is δ(x, y) = y+ 1.
(2) Calculate the following integrals. You will need them for Question 3:
(a) Z2
x=−2
√4−x2dx
(b) Z2
x=−2
x√4−x2dx
(c) Zcos2(x)dx
(Answers: 2π, 0, x
2+1
4sin(2x) + C.)
(3) Let Sbe the solid which is bounded below by the plane z= 0, on the sides by the
surface x2+y2= 4, and above by the plane z= 3 −x. (See the picture in #32, page
815 in Hass-Weir-Thomas.) Suppose the density is constant throughout S.
(a) In words, what would you call this solid? (There is no right or wrong answer to
this part of the question. I’m fishing for suggestions. I may use this example in
the future, and I don’t know what people call it.)
(b) Without doing any calculations, what is ¯y?
(c) Set up an integral in rectangular coordinates to find M, the total mass of S.
Evaluate it to find M. (Use your work from Question 2!)
(d) Change your integral to cylindrical coordinates. Evaluate this integral. You
should get the same answer as in part (b). Which integral was easier?
(e) Find Myz . (This is easier in cylindrical coordinates.)
(f) Find Mxy . (This is easier in cylindrical coordinates.)
(g) Find the center of mass.
(4) (a) Change the point (2, π/4,1) in cylindrical coordinates into rectangular coordi-
nates. Plot the point.
(b) Change the point (1, π, e) in cylindrical coordinates into rectangular coordinates.
Plot the point.
(c) Change the point (1,−1,4) in rectangular coordinates into cylindrical coordi-
nates. Plot the point.
(d) Change the point (3,4,5) in rectangular coordinates into cylindrical coordinates.
Plot the point.
(5) (a) Change the point (3, π, 0) in spherical coordinates (in the order (ρ, ϕ, θ )) into
rectangular coordinates. Plot the point.
(b) Change the point (2, π/3, π /4) in spherical coordinates (in the order (ρ, ϕ, θ))
into rectangular coordinates. Plot the point.
(c) Change the point (2, π/4, π /3) in spherical coordinates (in the order (ρ, ϕ, θ))
into rectangular coordinates. Plot the point.
1
