Math 209 , Final Exam Practice QuestionsMultiple Choice Questions

1. If C is the path given by r(t) = cos50(π2(1− t))i + sin100(πt)j + t1000k, 0 ≤ t ≤ 1 and

F(x, y, z) = (yexy + z cos(xz) + 2x)i + (y + xexy)j + (x cos(xz))k,

then

∫C

F · dr equals

(A) −2e (B) −1− sin(1) (C) 0

(D) 1 + sin(1) (E) 2e

(Answer) D

2. For every differentiable function f = f(x, y, z) and differentiable 3-dimensional vector fieldF = F(x, y, z) the vector field Curl(fF) equals

(A) f Curl(F)− (5f)× F (B)f Curl(F) + (5f)× F (C) (5f · F)5 f

(D) div (F)5 f (E) (5f · F)F

(Answer) B

3. The volume of the solid region inside both the sphere x2 + y2 + z2 = 6 and the paraboloidz = x2 + y2 equals

(A) 2π(2√

6− 9) (B) 2π(6√

6− 11)

(C)2π

3(6√

6− 11) (D)2π

3(2√

6− 11) (E)2π

3(2√

6− 9)

(Answer) C

4. A thin wire is bent in the shape of a semicircle x2 +y2 = 4, x ≤ 0. If the density is a constantk, then the x-coordinate of the center of mass is:

(A) − 4π

(B) − 3π

(C) − 2π

(D) − 1π

(E) 0

(Answer) A

5. The iterated integral

∫ 8

0

∫ 2

y13

ex4

dxdy equals

(A) 14e16 (B) 1

4(e16 + 1) (C) 1

4(e16 − 1) (D) 1

2(e8 − 1)

(E) 12e8

(Answer) C

6. A lamina occupies the region inside x2+y2 = 2y but outside x2+y2 = 1. If the density at anypoint is inversely proportional to its distance from the origin, with constant of proportionalityK, then the y-coordinate of the center of mass equals:

(A) 0 (B)√3√

3−π3

(C)√3

2(√3−π

3)

(D) 33+ π√

3

(E) 3√3

2(3√3+π)

(Answer) C

Long Answer Questions

1. Use the given transformation to evaluate the integral..∫ ∫R

(x− 10y) dA, where R is the triangle region with vertices (0, 0), (9, 1), (1, 9). and

x = 9u+ v, y = u+ 9v.

(Answer) −1200

2. Verify Stokes’ Theorem for F(x, y, z) = y2i+ xj+ z2k, and surface S given by the part of theparaboloid z = x2 + y2 that lies below the plane z = 1, oriented upward.

(Answer)

∫C

F · dr =

∫ ∫S

Curl(F).ds = π

3. Use the Divergence Theorem to find the upward flux of

F(x, y, z) = (6x+ y2)i + (5x2y +5

3y3 − x3)j + (8z + 1)k

through the surface S given by the part of the cone z = 2(1−√x2 + y2) that lies above the

xy-plane.

Hint: Consider also a surface S1 such that S and S1 together enclose a solid.

(Answer) 343π

4. Evaluate the surface integral

∫ ∫S

yz ds where S is the portion of the cone z =√x2 + y2

that lies within the first octant and inside the cylinder x2 + y2 = 1.

(Answer)√24