Math 2210-1
Homework 7
Due Wednesday July 14
Show all work. Please box
√ your answers. Be sure to write in complete sentences when appropriate. Also,
I prefer exact answers like 2 instead of 1.414. Note that a symbol indicates that graph paper might be
useful for that problem.
Double Integrals in Polar Coordinates
1.
For each of the regions R below, set up
Z
f dA as an iterated integral in polar coordinates.
R
y
2
y
2
1
x
-2
-1
0
(a)
1
1
(b)
y
3
(c)
-3
y
x
-0.5
(d)
0.5
-0.5
Sketch the regions over which the following integrals are computed.
(a)
Z
2π
Z
π
Z
π/3
Z
4
Z
π/4
Z
π/2
0
(b)
2
Z
1
3
(e)
Z
1
f (r, θ)r dr dθ
0
Z
3π/2
f (r, θ)r dθ dr
3π/4
0
(f)
f (r, θ)r dr dθ
0
π/6
(d)
f (r, θ)r dr dθ
1
π/2
(c)
Z
Z
1/ cos θ
Z
2/ sin θ
f (r, θ)r dr dθ
0
π/4
0
f (r, θ)r dr dθ
2
0.5
3 x
-3
2.
x
2
(g)
Z
4
0
3.
Z
π/2
f (r, θ)r dθ dr
−π/2
Evaluate the integrals below over the region indicated
(a)
Z
sin(x2 + y 2 ) dA where R is the disc of radius 2 centered at the origin.
Z
(x2 − y 2 ) dA where R is the first quadrant region between the circles of radius 1 and radius 2.
R
(b)
R
4.
Change the following integrals to polar coordinates and evaluate.
(a)
(b)
Z
0
Z
1−x2
x dy dx
√
−1 − 1−x2
Z √2 Z √4−y2
xy dx dy
y
0
5.
√
A disk of radius 5 cm has density 10 g/cm2 at its center, has density 0 at its edge, and its density
is a linear function of the distance from the center. Find the mass of the disk.
Applications of Double Integrals
6. Find the mass m and center of mass (x̄, ȳ) of the lamina bounded by y = 0, y = sin x, 0 ≤ x ≤ π with
the density given by δ(x, y) = y.
7. Find the moments of intertia Ix , Iy , Iz for the lamina bounded by the triangle with vertices (0,0), (0, a),
(a, a), with the density given by δ(x, y) = x2 + y 2 .
Surface Area
8.
Make a sketch of the following surfaces and find the surface area:
(a) The part of the plane 3x − 2y + 6z = 12 that is bounded by the planes x = 0, y = 0, and
3x + 2y = 12.
p
(b) The part of the surface z = 4 − y 2 in the first octant that is directly above the circle x2 + y 2 = 4
in the xy-plane.
Triple Integrals (cartesian Coordinates)
9.
Find the three-variable integral of the given functions over the given regions.
(a) f (x, y, z) = x2 + 5y 2 − z, W is the rectangular box 0 ≤ x ≤ 2, −1 ≤ y ≤ 1, 2 ≤ z ≤ 3
(b) g(x, y, z) = sin x cos(y + z), W is the rectangular box 0 ≤ x ≤ π, 0 ≤ y ≤ π, 0 ≤ z ≤ π
(c) h(x, y, z) = ax + by + cz, W is the rectangular box 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1
10.
For the following problems, describe or sketch the region of integration for the triple integrals.
the limits do not make sense, explain why.
Z 6 Z 3−x/2 Z 6−x−2y
(a)
f (x, y, z) dz dy dx
0
(b)
0
Z
1
Z
1
0
(c)
0
Z
0
Z
0
0
x
Z
x
0
zZ x
0
f (x, y, z) dz dy dx
f (x, y, z) dz dy dx
If
(d)
Z
3
Z
3
Z
1
Z
1
0
(e)
Z
x+y
Z
2−x
√
9−y 2
1
0
(g)
0
−
1
(f)
Z
−1
3
√
Z
y
Z
3
f (x, y, z) dz dx dy
x2 +y 2
f (x, y, z) dz dx dy
0
0
Z
Z
f (x, y, z) dz dy dx
√ 2 2
2 Z
0
√
0
1−x
2−x −y
f (x, y, z) dz dy dx
0
11. Find the average value of the sum of the squares of three numbers x, y, z where each number is between
0 and 2.
Integrals in Cylindrical and Spherical Coordinates
12.
Evaluate the indicated integrals in cylindrical coordinates.
(a)
(b)
Z
ZW
x2 + y 2 + z 2 dV where W is the region 0 ≤ r ≤ 4, π/4 ≤ θ ≤ 3π/4, −1 ≤ z ≤ 1.
sin(x2 + y 2 ) dV where W is the solid cylinder with height 4 and with base of radius 1 centered
W
on the z-axis at z = −1.
13.
Evaluate the indicated integrals in spherical coordinates.
(a)
Z
(b)
Z
W
W
14.
1
p
dV where W is the bottom half of the sphere of radius 5 centered at the origin.
2
x + y2 + z 2
sin φ dV where W is the region 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/4, 1 ≤ ρ ≤ 2.
Sketch the region over which the integration is being performed:
Z
π/2
0
15.
π
π/2
Z
1
f (ρ, φ, θ)ρ2 sin φ dρ dφ dθ.
0
Evaluate the following integrals:
1
(a)
Z
1
(b)
Z
0
0
16.
Z
Z
√
1−x2
√
Z
√
1−x2 −z 2
√
− 1−x2 −z 2
− 1−x2
√
1−x2
1 Z
Z
−1
√
− 1−x2
1
p
dy dz dx.
2
x + y2 + z 2
1
p
dy dx dz.
x2 + y 2
Find the volume that remains after a cylindrical hole of radius a is bored through a sphere of radius
R, where 0 < a < R, passing through the center of the sphere along the pole.