Winter 2017, MATH 215 Calculus III, Final Exam
4/24/2017, 10:30 am -12:30 pm (120 minutes)
• Your name:
• Circle your section and write your Lab time:
Section Time Professor GSI Lab Time (e.g. Th 10-11)
20 9–10 Sema Gunturkun Alex Leaf
30 10–11 Mattias Jonsson Robert Cochrane
40 11–12 Sumedha Ratnayake Harry Lee
50 12–1 Sumedha Ratnayake Deshin Finlay
60 1–2 Yueh-Ju Lin Rebecca Sodervick
70 2-3 Yueh-Ju Lin Jacob Haley
Instructions:
• This examination booklet contains 9 problems.
• If you want extra space, write on the back..
• DO NOT remove any sheets or the staple from the exam booklet.
• The formula sheet is not collected back and not graded.
• This is a closed book exam. Electronic devices, calculators, and note-cards are not allowed.
• Show your work and explain clearly.
Problem Your score Maximum score
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
Total 90
1. (10 points) Evaluate the double integral
ˆ ˆ
xy dA,
D
where D is the triangular region with vertices at (0, 0), (0, 2), and (1, 1).
Value of the integral =
Extra space:
2. (10 points) Two planes are said to be orthogonal if their normal vectors are orthogonal. Find the
equation of a plane that is orthogonal to the plane x−2y+z = 3 and contains the line with parametric
equation (x, y, z) = (1 + 2t, 2 − t, −1 + 2t).
Equation of the plane:
Extra space:
3. (10 points) Let F = xzi + yj + xk. Find the flux
ˆ
F · n dS
out of the surface of the tetrahedron with vertices at (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1). Here n
is the unit outward normal as usual.
Flux =
Extra space:
4. Below F(x, y, z) is a vector field and f (x, y, z) is scalar valued.
(a) (5 points) Find f such that F = ∇f for F = z cos yi − xz sin yj + x cos yk.
(b) (5 points) Verify that there is no f with F = ∇f for F = z cos yi + xz sin yj + x cos yk.
(a): f =
(b) There is no such f because:
Extra space:
5. Consider the solid sphere x2 + y 2 + z 2 ≤ 1.
(a) (6 points) Assume that the density (mass per unit volume) is equal to z + 1. Find the z-
coordinate of the center of mass of the solid sphere.
(b) (4 points) Assume that the density is equal to x + 1. Find the x-coordinate of the center of
mass of the solid sphere.
(a): z-coordinate is :
(b): x-coordinate is::
Extra space:
6. Consider the paraboloid surface P given by z = 1 − (x2 + y 2 ), 0 ≤ x2 + y 2 ≤ 1.
(a) (5 points) Find the area of the surface P .
(b) (5 points) Evaluate the surface integral
ˆ ˆ
F · n dS,
P
where P is the same paraboloid surface and F = x2 i + y2 j + zk. As usual, n is the unit normal
to the surface (in the upward direction) and dS is the area element. The disc at the bottom is
not included in the surface.
(a): Surface area of P :
(b): Flux:
Extra space:
7. (10 points) The following four plots show vector fields or flows. It is assumed that the vector field
has no z-component and that the flow is the same in all planes parallel to the x-y plane. Therefore
the only component of the curl that can be nonzero is the z-component. The z-axis is perpendicular
to the plane of the paper and pointing towards the sky.
For each of the points A to G, figure out whether the z-component of the curl is negative, zero, or
positive. Read the titles of each plot carefully. It is important that D is located exactly on the
midline, where the flow flips direction. Likewise, it is significant that F is the center.
Hint: consider loops of an appropriate shape and think in terms of Stokes’ theorem.
1.0 Constant flow 1.0 Constant direction, increasing speed
0.5 0.5
0.0 A 0.0 B
0.5 0.5
1.0 1.0 0.5 0.0 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0
1.0 Constant speed, direction flip in midline 1.5 Circling, speed =1/r
1.0
0.5 C
0.5
0.0 D 0.0 F G
0.5
0.5 E
1.0
1.0 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
(answer zero, positive, or negative in each case)
A:
B:
C:
D:
E:
F:
G:
Extra space:
8. Consider the line integral ˆ
−y x
dx + dy,
C x + y2
2 x + y2
2
where C is assumed to be a simple closed curve with positive orientation.
(a) (3 points) Calculate the line integral for the curve x2 + y 2 = 1.
(b) (3 points) Evaluate and simplify
∂ x ∂ y
+ .
∂x x + y2
2 ∂y x + y2
2
x2 y2
(c) (4 points) Calculate the line integral for the ellipse 16
+ 25
= 1.
(a): Line integral =
(b): Simplified value =
(c): Line integral =
Extra space:
9. The plane 3x + 6y + 2z = 7 slices the spherical surface x2 + y 2 + z 2 = 4 into two parts.
(a) (2 points) Find the distance of the plane from the center of the sphere.
(b) (8 points) Find the surface area of each part of the sphere.
(a) The distance is:
(b) The surface areas of the two parts are:
Extra space:
