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MATH 261 - SPRING 2001
FINAL EXAM
Name Instructor
Signature Recitation Instructor
Div. Sect. No.
FINAL EXAM INSTRUCTIONS
- You must use a #2 pencil on the mark-sense sheet (answer sheet).
- On the mark-sense sheet, fill in the instructor’s name and the course number.
- Fill in your name and student identification number and blacken in the appropriate
spaces.
- Mark in your division and section number of your class. For example, for division 02,
section 03, fill in 0203 and blacken the corresponding circles, including the circles for
the zeros. (If you do not know your division and section number ask your instructor.)
- Sign the mark-sense sheet.
- Fill in the information above and fill in your name on each of the question sheets.
- There are 20 questions, each worth 10 points. Blacken in your choice of the correct an-
swer in the spaces provided for questions 1–20. Do all your work on the question sheets.
Turn in both the mark-sense sheets and the question sheets when you are finished.
- No partial credit will be given, but if you show your work on the question sheets it
may be considered if your grade is on the borderline.
- Calculators are not allowed. NO BOOKS OR PAPERS ARE ALLOWED. Use the
back of the test pages for scratch paper.
- Give the equation of a plane containing the points (1, 0 , 0) and (1, 2 , 1) and parallel to
the line whose equations are x = y, z = 0.
A. x + y = 1
B. x − y = 1
C. x − y + z = 1
D. x−y +2z = 1
E. x + y − z = 1
- Let ~r(t) = e
t cos t~i + e
t sin t~j be parametric equations of a curve C. Find the length of
C from t = 0 to t = π.
A.
2 e
π
B.
2 (e
π −e
−π )
C.
2 (e
π − 1)
D.
2 (e
π
+e
−π
)
E.
2 (e
π
- The maximum value of f (x, y) = 2x + 4y on the circle x
2
2 = 5 is:
A. 6
B. 8
C. 10
D. 4
E. 6
- The function f (x, y) = x
3
3
- 3xy has critical points (0, 0) and (− 1 , −1). These
critical points are
A. both maximum points
B. both minimum points
C. one minimum and one maximum point
D. one minimum and one saddle point
E. one maximum and one saddle point
- If the order of integration is reversed, which of the following integrals is equal to
√
π
0
π
y
2
(sin x
2
)dx dy?
A.
π
0
π
√
x
(sin x
2
)dy dx
B.
π
0
√
x
0
(sin x
2
)dy dx
C.
√
π
√
x
π
0
(sin x
2
)dy dx
D.
π
0
√
π
x
(sin x
2
)dy dx
E.
√
π
0
π
x
(sin x
2
)dy dx
- If R is the region in the xy-plane inside the circle x
2
2
= 1 and above the line y = x,
then
R
x dA expressed in polar coordinates is:
A.
∫ 3 π
2
π
2
1
0
r cos θdr dθ
B.
∫ 3 π
2
π
2
1
0
r
2
cos θdr dθ
C.
∫ 5 π
4
π
4
r cos θ
0
r
2
cos θdr dθ
D.
∫ 5 π
4
π
4
r
2
0
r
2
cos θdr dθ
E.
∫ 5 π
4
π
4
1
0
r
2
cos θdr dθ
- If D is the solid region in the first octant bounded above by the paraboloid z =
1 − x
2
− y
2
and below by the xy-plane, the volume of D is:
A.
π
B.
π
C.
π
D.
π
E.
π
- The integral
√
2
0
√
2 −x
2
−
√
2 −x
2
8 −x
2 −y
2
3 x
2 +3y
2
xyz dz dy dx in cylindrical coordinates is:
A.
∫ π
2
0
√
2
0
√
8 −r
2
√
3 r
r
3
z cos θ sin θdz dr dθ
B.
∫ π
2
0
2
0
√
8 −r
2
√
3 r
r
2
z cos θ sin θdz dr dθ
C.
∫ π
2
−
π
2
√
2
0
√
8 −r
2
√
3 r
r
3
z cos θ sin θdz dr dθ
D.
∫ π
2
−
π
2
2
0
√
8 −r
2
√
3 r
r
3
z cos θ sin θdz dr dθ
E.
∫ π
2
−
π
2
2
0
√
8 −r
2
√
3 r
r
2
z cos θ sin θdz dr dθ
- The mass of an object occupying the region bounded above by the plane z = 2 and
below by the upper nappe of the cone z
2
= x
2
2
with mass density at each point
equal to x
2
2
2 is given by:
A.
2 π
0
∫ π
4
0
2
0
ρ
2
sin φ dρ dφ dθ
B.
2 π
0
∫ π
4
0
2 sec φ
0
ρ
4
sin φ dρ dφ dθ
C.
2 π
0
∫ π
4
0
2
0
ρ
4
sin φ dρ dφ dθ
D.
2 π
0
∫ π
4
−
π
4
2 sec φ
0
ρ
3
sin
2
φ dρ dφ dθ
E.
2 π
0
∫ π
4
0
2 sec φ
0
ρ
3
sin
2
φ dρ dφdθ
- Let
F (x, y, z) = (z + y
2 )
i + 2xy~j + (x + y)
k. Find the curl(
F ) at the point (1, 1 , 1).
A.
i +
j +
k
B.
j
C.
i
D. −
i
E. −
j
- If
F (x, y) = (2xe
y
i + (x
2 e
y )
j is a conservative vector field, that is
F (x, y) = grad f (x, y) for some function f , and C is any smooth curve from (0, 0) to
(1, 1) then
C
F · d~r =
A. 0
B. 1
C. 2 e
D. e + 1
E. 2 e + 1
- Let R be the region bounded by the curve x = y−y
2 and the y-axis. If C is the boundery
of the region R oriented counterclockwise, then
C
(e
2 x
2
)dx + (14xy + y
2
)dy =
A. 0
B.
C.
D. 1
E. 2
- Compute the surface integral
Σ
4 z dS, where Σ is the part of the sphere x
2
+y
2
+z
2
=
10 which lies above the plane z = 1.
A. 4
10 π
B. 36
10 π
C. 40
10 π
D. 19 π
E. 99 π
- Let D be the solid bounded by the cone z =
x
2
2 and the plane z = 1 whose
boundary surface Σ is oriented by the unit normal ~n directed outward from D. If
F (x, y, z) = (8xz)
i + (z
3 e
−x )
j + (y cos x)
k then
Σ
F · ~n dS =
A. 0
B. π
C.
π
D. 2 π
E.
16 π
