Fall 2014
Math 265 - Practice Exam 3, Integrals
Problem 1 Find the area of the region S in the first quadrant enclosed by
the curve r = 3 cos 2θ (in polar coordinates).
Problem 2 a) Find the volume of the solid in the first octant bounded by
x + 2y + 3z = 6.
b) Find the mass of the solid from part a) if the density is given by δ(x, y, z) =
y.
c) Find the x-coordinate of the center of mass of the solid above.
Problem 3 A function f (x, y) is integrated over the square S with vertices
A = (0, 0), B = (2, 0), C = (2, 2), and D = (0, 2). Denote a triangle by its
three vertices. With M being the point (1, 1), it is given that
ZZZ
ZZZ
f (x, y) dV = 10,
f (x, y) dV = 4,
ZZZ S
Z Z ZABC
f (x, y) dV = 7
f (x, y) dV = 3.
BCD
BCM
Determine the integral of f (x, y) over the triangle AM D.
Problem 4 Rewrite the integral below with the order of integration dz dy dx.
Z
2/3 Z 1−3y/2 Z 2−3y−2z
I=
f (x, y, z) dx dz dy
0
Problem 5
0
0
Consider the solid S bounded by the coordinate planes and by the planes
z = 3 − y, z = 2, and 4 − x = z.
a) Find inequalities for x, y, z describing S (for each of the given equations,
you have to pick ≤ or ≥).
b) Set up an integral for the volume of S in order dz dy dx.
c) Set up an integral for the volume of S in order dx dy dz.
d) Find the volume of S using the integral of your choice (one is easier than
the other one!).
Problem 6 A solid is defined by x2 + y 2 + z 2 ≤ 4 and 0 ≤ x ≤ y.
a) Set up the triple integral for its volume in cylindrical coordinates (r, θ, z).
b) Set up the triple integral for its volume in spherical coordinates (ρ, φ, θ).
c) Find the mass of the solid if the density is δ = 1/ρ.
Problem 7 Let R be the region defined by 0 ≤ 2(x+y) ≤ π and (6x−2y)2 ≤
1 (this is indeed a finite region). Evaluate the double integral
Z Z
cos(x + y)e3x−y dA
R
using the transformation u = x + y and v = 3x − y.