Math 3C (21945) HW #8
Due at the beginning of lecture on Thursday, April 23rd.
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neatly and legibly. You cannot simply write the correct answer to demonstrate your mathematical
understanding.
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must be stapled. No late homework will be accepted without my express permission. You may receive a
✗ if these guidelines are not followed.
Good luck!
Evaluate the integral.
4
3
8
1)
xyz dx dy dz
0
0
0
∫ ∫ ∫
2)
3
∫ ∫
0
3)
4(1 - z/3)
0
1
7(1 - y/4 - z/3)
dx dy dz
2)
0
1
∫ ∫ ∫
-1
∫
1)
0
4
(x2 + y 2 + z2) dx dy dz
3)
0
Find the volume of the indicated region.
4) the region bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and x + z = 9
4)
5) the region common to the interiors of the cylinders x2 + y 2 = 36 and x2 + z2 = 36
5)
Evaluate the integral by changing the order of integration in an appropriate way.
1
4
4
x sin z dz dy dx
6)
z
0
0
y
∫ ∫ ∫
7)
7
0
8)
7
∫ ∫ ∫
y
8
0
2
∫ ∫ ∫
0
1
!
6)
sin z sin x dz dx dy
x
7)
3
ey dy dz dx
z
8)
1
x/8
Solve the problem.
9) Set up the triple integral for the volume of the sphere ! = 3 in spherical coordinates.
1
9)
10) Set up the triple integral for the volume of the sphere ! = 4 in cylindrical coordinates.
10)
#
11) Let D be the region that is bounded below by the cone " =
and above by the sphere
4
11)
! = 5. Set up the triple integral for the volume of D in cylindrical coordinates.
Evaluate the spherical coordinate integral.
!
!
5 sin "
12)
#2 sin " d# d" d$
0
0
0
∫ ∫ ∫
13)
∫
8!
0
!
∫ ∫
0
(1 - cos ")/2
12)
#2 sin " d# d" d$
13)
0
Use spherical coordinates to find the volume of the indicated region.
14) the region bounded above by the sphere x2 + y 2 + z2 = 64 and below by the cone
14)
x2 + y 2
z=
#
#
15) the region inside the solid sphere ! ≤ 3 that lies between the cones " =
and " =
3
2
16) the region that lies inside the sphere x2 + y 2 + z2 = 49 and outside the cylinder
x2 + y 2 = 36
15)
16)
Change the order of integration and evaluate the integral.
!
(1 - cos ")/2
10!
17)
#2 sin " d$ d# d"
0
0
0
17)
Solve the problem.
18) Find the center of mass of the rectangular solid of density !(x, y, z) = xyz defined by
0 ≤ x ≤ 9, 0 ≤ y ≤ 10, 0 ≤ z ≤ 3.
18)
∫ ∫
∫
19) Find the center of mass of the region of density !(x, y, z) =
1
bounded by the
4 - x2 - y 2
19)
paraboloid z = 4 - x2 - y 2 and the xy-plane.
Use the given transformation to evaluate the integral.
20) x = 8u, y = 4v, z = 7w;
∫∫∫
20)
x2y 2 dx dy dz,
R
2
2
2
where R is the interior of the ellipsoid x + y + z = 1
64 16
49
2
21) u = 2x + y - z, v = -x + y + z, w = -x + y + 2z;
∫∫∫
21)
(2x + y - z) dx dy dz,
R
where R is the parallelepiped bounded by the planes 2x + y - z = 2, 2x + y - z = 4,
-x + y + z = 8, -x + y + z = 10, -x + y + 2z = 7, -x + y + 2z = 10
3
Answer Key
Testname: M3C_HW_8
1) 1152
2) 14
3) 48
4) 81!
5) 1152
1 - cos 4
6)
2
7) 2(1 - cos 7)
8
8) ln 2 (e - 1)
3
9)
∫
2!
!
0
∫ ∫
10)
∫
0
2!
∫
∫ ∫
0
2!
0
∫
"2 sin # d" d# d$
0
4
0
11)
3
-
16 - r2
r dz dr d$
16 - r2
5/ 2
0
∫
25 - r2
r dz dr d$
r
125 2
12)
!
8
13)
4
!
3
14)
512
!(2 3
2)
15) 9!
16)
4(343 - 133/2)!
3
17)
5
!
3
18) x = 6, y =
20
,z=2
3
19) x = 0, y = 0, z = 1
2
20)
128
!
15
21) 12
4