Math 209 , Final Exam Practice Questions
Multiple Choice Questions
1. If C is the path given by r(t) = cos50 ( π2 (1 − t))i + sin100 (πt)j + t1000 k, 0 ≤ t ≤ 1 and
F(x, y, z) = (yexy + z cos(xz) + 2x)i + (y + xexy )j + (x cos(xz))k,
Z
F · dr equals
then
C
(A) −2e
(B) −1 − sin(1)
(D) 1 + sin(1)
(C) 0
(E) 2e
(Answer) D
2. For every differentiable function f = f (x, y, z) and differentiable 3-dimensional vector field
F = F(x, y, z) the vector field Curl(f F) equals
(A) f Curl(F) − (5f ) × F
(D) div (F) 5 f
(B)f Curl(F) + (5f ) × F
(C) (5f · F) 5 f
(E) (5f · F)F
(Answer) B
3. The volume of the solid region inside both the sphere x2 + y 2 + z 2 = 6 and the paraboloid
z = x2 + y 2 equals
√
√
(A) 2π(2 6 − 9)
(B) 2π(6 6 − 11)
(C)
2π √
(6 6 − 11)
3
(Answer) C
(D)
2π √
(2 6 − 11)
3
(E)
2π √
(2 6 − 9)
3
4. A thin wire is bent in the shape of a semicircle x2 + y 2 = 4, x ≤ 0. If the density is a constant
k, then the x-coordinate of the center of mass is:
(A) − π4
(B) − π3
(C) − π2
(D) − π1
(E) 0
(Answer) A
Z
8
Z
5. The iterated integral
1
y3
0
(A)
1 16
e
4
2
(B)
4
ex dxdy equals
1 16
(e
4
+ 1)
(C)
1 16
(e
4
− 1)
(D)
1 8
(e
2
− 1)
(E) 21 e8
(Answer) C
6. A lamina occupies the region inside x2 + y 2 = 2y but outside x2 + y 2 = 1. If the density at any
point is inversely proportional to its distance from the origin, with constant of proportionality
K, then the y-coordinate of the center of mass equals:
(A) 0
(E)
√
3√ 3
2(3 3+π)
(Answer) C
(B)
√
√ 3π
3− 3
(C)
√
√ 3π
2( 3− 3 )
(D)
3
3+ √π
3
Long Answer Questions
1. Z
UseZ the given transformation to evaluate the integral..
(x − 10y) dA, where R is the triangle region with vertices (0, 0), (9, 1), (1, 9). and
R
x = 9u + v, y = u + 9v.
(Answer) −1200
2. Verify Stokes’ Theorem for F(x, y, z) = y 2 i + xj + z 2 k, and surface S given by the part of the
paraboloid z = x2 + y 2 that lies below the plane z = 1, oriented upward.
Z
Z Z
F · dr =
(Answer)
C
Curl(F).ds = π
S
3. Use the Divergence Theorem to find the upward flux of
5
F(x, y, z) = (6x + y 2 )i + (5x2 y + y 3 − x3 )j + (8z + 1)k
3
p
through the surface S given by the part of the cone z = 2(1 − x2 + y 2 ) that lies above the
xy-plane.
Hint: Consider also a surface S1 such that S and S1 together enclose a solid.
(Answer)
34
π
3
Z Z
4. Evaluate the surface integral
yz ds where S is the portion of the cone z =
S
that lies within the first octant and inside the cylinder x2 + y 2 = 1.
√
(Answer)
2
4
p
x2 + y 2