2415 - Quiz 6.tst
Write an equivalent double integral with the order of integration reversed.
1)
2
0
4
y2
4y dx dy∫∫
A)
4
0
x
2
4y dy dx∫∫ B)
4
0
x
0
4y dy dx∫∫
C)
2
0
x
0
4y dy dx∫∫ D)
2
0
x
2
4y dy dx∫∫
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
2) The parabola y = x2 and the line y = 3x + 28
A)
7
-4
3x + 28
x2
dy dx∫∫ B)
7
0
3x + 28
x2
dy dx∫∫
C)
7
0
3x + x2
28
dy dx∫∫ D)
7
-4
3x + 28 - x2
0
dx dy∫∫
Evaluate the integral.
3)
π/8
0
cos 4x
0
sin 4x dy dx∫∫
A) π
16
B) π
8
C) 1
16
D) 1
8
Integrate the function f over the given region.
4) f(x, y) = xy over the triangular region with vertices (0, 0), (9, 0), and (0, 7)
A) 189
8
B) 21
8
C) 147
8
D) 1323
8
Reverse the order of integration and then evaluate the integral.
5)
8
0
1
x/8
ey3 dy dx∫∫
A) 8
3
(e - 1) B) 8
3
(2e - 1) C) 4
3
(2e - 1) D) 4
3
(e - 1)
Find the volume of the indicated region.
6) the region bounded by the paraboloid z = 100 - x2 - y2 and the xy-plane
A) 10000
3
π B) 5000π C) 5000
3
π D) 2500π
Sumit Sarka
Find the center of mass of a thin plate covering the given region with the given density function.
7) The region bounded below by the parabola y = x2 and above by the line y = x + 2, with density ρ(x) = 6x2
A) x = 8
7
, y = 118
49
B) x = 531
70
, y = 18
5
C) x = 18
5
, y = 531
70
D) x = 9
8
, y = 131
64
Evaluate the integral.
8)
2
0
y2
1
z
2
yz dx dz dy∫∫∫
A) - 128
3
B) 23
3
C) 68
3
D) 4
3
Use cylindrical coordinates to find the volume of the indicated region.
9) the region enclosed by the cylinder x2 + y2 = 9 and the planes z = 0 and x + y + z = 6
A) 108π B) 81π C) 27π D) 54π
Evaluate the spherical coordinate integral.
10)
π/2
0
π/2
0
8 cos φ
0
ρ2 sin φ dρ dφ dθ∫∫∫
A) 64
3
π2 B) 256
9
π C) 64
3
π D) 256
9
π2
Evaluate the line integral of f(x,y) along the curve C.
1) f(x, y) = x2 + y2, C: y = 4x + 2, 0 ≤ x ≤ 3
A XXXXXXXXXXB XXXXXXXXXXC) 237 D XXXXXXXXXX
Evaluate the line integral along the curve C.
2)
C
(xz + y2)∫ ds, C is the curve r(t) = (-7 - 2t)i + tj - 2tk , 0 ≤ t ≤ 1
A) 26
3
B) 26 C) - 16
3
D) - 16
Evaluate the line integral.
3)
C
y2 dx + x dy∫ ; C is the curve x = t2 - 1, y = 6t, 0 ≤ t ≤ 1
A) 26 B) 14 C) 5
6
D) 72
Find the work done by F over the curve in the direction of increasing t.
4) F = xyi + 8j + 3xk; C: r(t) = cos 8ti + sin 8tj + tk, 0 ≤ t ≤ π
16
A) W = 25
3
B) W = 209
24
C) W = 0 D) W = 193
24
Find the potential function f for the field F.
5) F = (y - z)i + (x + 2y - z)j - (x + y)k
A) f(x, y, z) = xy + y2 - xz - yz + C B) f(x, y, z) = x + y2 - xz - yz + C
C) f(x, y, z) = x(y + y2) - xz - yz + C D) f(x, y, z) = xy + y2 - x - y + C
Evaluate the work done between point 1 and point 2 for the conservative field F.
6) F = 2xi + 2yj + 2zk; P1(2, 2, 5) , P2(3, 6, 9)
A) W = 159 B) W = 0 C) W = 93 D) W = - 93
Find the divergence of the field F.
7) F = -6x7i - 8xyj - 4xzk
A) -42x6 - 12x B) -54 C) -42x6 - 8y - 4z D) -42x6 - 12x - 54
Find curl F.
8) F(x, y, z) = 2xyzi + 8yzj + 5zk
A) -8yi + 2xyj + -2xzk B) -8y + 2xy - 2xz
C) 2yzi + 8zj + 5k D) 2yz + 8z + 5
Apply Greenʹs Theorem to evaluate the integral.
9)
C
(x + 2y)dx + 4xy dy; C the the triangle with vertices (0, 0), (4, 0) and (0, 4).
A) 80 B) 160
3
C) 80
3
D) 160
Using Greenʹs Theorem, compute the counterclockwise circulation of F around the closed curve C.
10) F = - 1
8(x2 + y2)4
i; C is the region defined by the polar coordinate inequalities 1 ≤ r ≤ 2 and 0 ≤ θ ≤ π
A) 127
448
B) 129
448
C) 0 D) - 127
448
Evaluate the surface integral of f over the surface S.
1) S is the cylinder y2 + z2 = 100, z ≥ 0 and 5 ≤ x ≤ 8; f(x, y, z) = z
A) 600 B) 60 C) 300 D) 2600
2) S is the hemisphere x2 + y2 + z2 = 7, z ≥ 0; f(x,y,z) = z2
A) 28
3
π B) 98
3
π C) 196π D) 49
3
π
3) S is the plane x + y + z = 1 above the rectangle 0 ≤ x ≤ 5 and 0 ≤ y ≤ 2; f(x,y,z) = 2z
A) 120 B) 80 3 C) - 50 3 D) -80
Find the flux of the vector field F across the surface S in the indicated direction.
4) F = 4xi + 4yj + zk; S is portion of the plane x + y + z = 3 for which 0 ≤ x ≤ 5 and 0 ≤ y ≤ 2; (Oriented upward)
A) 270 B) - 75 C) 135 D) 120
Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
5) F = x2i + y2j + zk; D: the solid cube cut by the coordinate planes and the planes x = 2, y = 2, and z = 2
A) 34 B) 40 C) 24 D) 16
6) F = 2xy2i + 2x2yj + 2xyk ; D: the region cut from the solid cylinder x2 + y2 ≤ 4 by the planes z = 0 and z = 2
A) 128π B) 32π C) 64π D) 16π
7) F = 6x3i + 6y3j + 6z3k ; D: the thick sphere 9 ≤ x2 + y2 + z2 ≤ 16
A) 666π B) 56,232π C) 56232
5
π D) 1400π
Use Stokesʹ Theorem to calculate
S
(curl F) ∙ n dS∫∫
8) F = 5yi - 6xj + 2z3k ; C: the portion of the plane 6x + 7y + 4z = 6 in the first quadrant
A) - 33
7
B) 0 C) -11 D) 33
7
Use Stokesʹ Theorem to calculate
S
(curl F) ∙ n dS∫∫ .
9) F = 5yi + 7xj + z3k; C: the counterclockwise path around the perimeter of the triangle in the x-y plane formed
from the x-axis, y-axis , and the line y = 5 - 2x (Hint: n = k)
A) 25
2
B) 5
2
C) 25 D) - 25
Use Stokesʹs Theorem to calculate
S
(curl F) ∙ n dS∫∫ .
10) F = (8 - y)i + (5 + x)j + z2k; S is the upper hemisphere of x2 + y2 + z2 = 16
A) 32π B) 64π C) -4π D) 2π