MATH 251 Spring 2021
EXAM III - VERSION B
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SECTION NUMBER:
UIN:
DIRECTIONS:
1. No notes, no calculators, no computers/laptops, no assistance of any kind may be used during this exam.
2. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your exam will be collected
and you will receive a zero.
3. In Part 1, mark the correct choice on your ScanTron using a No. 2 pencil. The ScanTron will not be returned,
therefore for your own records, also record your choices on your exam! Each problem is worth 4 points.
4. In Part 2, present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate
your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of
the work leading up to it.
5. Be sure to write your name, section number and version letter of the exam on the ScanTron form.
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Question Type
Points Awarded
Points
Multiple Choice
50
Free Response
50
Total
100
1
PART I: Multiple Choice. 4 points each.
1. Find the Jacobian of the transformation x = eu sin v and y = e2 cos v.
(a) e2u cos2 v
(b) −e2u
(c) −e2u sin2 v
(d) *−e2u
(e) e−2u
2. Consider the equation z =
(a) * φ =
p
3x2 + 3y 2 . Which of the following is true about φ in spherical coordinates?
π
6
(b) φ =
π
3
(c) φ =
π
4
(d) φ =
2π
3
(e) φ =
5π
6
(f) None of these
ZZZ
x dV , where E is the solid in the first octant that lies below the paraboloid z = x2 + y 2 , above
3. Compute
E
the xy plane, and inside the cylinder x2 + y 2 = 1.
(a)
π
8
(b)
π
2
(c) *
1
5
(d) 0
(e)
1
4
(f) None of these
2
4. Find the volume of the solid that lies under the plane z = y and above the region in the xy plane bounded by
y = x2 and x = y 2 .
(a)
1
15
(b) *
3
20
(c)
7
54
(d)
2
15
(e)
3
10
(f) None of these
5. Which of the following is a correct set up for the volume of the region that lies above the xy-plane and lies within
both the cylinder x2 + y 2 = 4 and the sphere x2 + y 2 + z 2 = 36?
Z 2π Z 2 Z √36−r2
(a) *
dz r dr dθ
0
0
0
Z π Z 2 Z √36−r2
dz r dr dθ
(b)
0
0
0
Z 2π Z π Z 6
(c)
0
0
ρ2 sin(φ)dρ dφ dθ
0
Z 2π Z 2 Z 6
(d)
dz r dr dθ
0
0
0
Z 2π Z π/2 Z 6
(e)
0
0
ρ2 sin(φ)dρ dφ dθ
0
(f) None of these
ZZZ
yz dV where E is the solid hemisphere x2 + y 2 + z 2 ≤ 9, z ≥ 0. Which of the following integrals is
6. Consider
E
the result after converting to spherical coordinates?
Z 2π Z π/2 Z 3
(a)
ρ2 sin2 (φ) cos(φ) sin(θ) dρ dφ dθ
0
0
0
Z 2π Z π Z 3
(b)
0
0
ρ4 sin2 (φ) cos(φ) sin(θ) dρ dφ dθ
0
Z π/2 Z π/2 Z 9
(c)
0
0
Z 2π Z π Z 3
(d)
0
ρ4 sin2 (φ) cos(φ) sin(θ) dρ dφ dθ
0
0
ρ2 sin2 (φ) cos(φ) sin(θ) dρ dφ dθ
0
Z 2π Z π/2 Z 3
(e) *
0
0
ρ4 sin2 (φ) cos(φ) sin(θ) dρ dφ dθ
0
3
7. Findpthe volume of the solid E that lies within the sphere x2 + y 2 + z 2 = 4, above the xy -plane and below the cone
z = x2 + y 2 using spherical coordinates.
√
8 2π
3
√
64 4π
(b)
3
√
4 2π
(c)
3
(a) *
(d)
16π
3
(e)
√
16π − 8 2
3
(f) None of these
ZZZ
xy dV where E is region bounded by x = 4y 2 + 4z 2 and x = 16. Which of the following is true?
8. Consider
E
Z 2π Z 4 Z 16
ZZZ
(a)
xy dV =
E
0
x r2 sin θ dx dr dθ
4r 2
0
Z 2π Z 2 Z 16
ZZZ
(b) *
xy dV =
E
0
0
Z 2π Z 2 Z 16
ZZZ
(c)
xy dV =
E
0
0
xy dV =
E
0
0
(e)
xy dV =
E
0
0
x r3 sin θ dx dr dθ
4r 2
Z 2π Z 4 Z 16
ZZZ
r3 sin θ cos θ dx dr dθ
4r 2
Z 2π Z 4 Z 16
ZZZ
(d)
x r2 cos θ dx dr dθ
4r 2
x r3 cos θ dx dr dθ
4r 2
9. Write the equation z = x2 − y 2 in spherical coordinates.
(a) cos φ = ρ sin2 φ sin2 θ − ρ sin2 φ cos2 θ
(b) * cos φ = ρ sin2 φ cos2 θ − ρ sin2 φ sin2 θ
(c) cos φ = ρ sin2 φ cos2 φ − ρ sin2 φ sin2 φ
(d) cos φ = ρ2 sin2 φ cos2 θ − ρ2 sin2 φ sin2 θ
(e) cos φ = ρ2 sin2 φ cos2 φ − ρ2 sin2 φ sin2 φ
4
ZZ
x sin y dA, where R is the region bounded by y = 0, y = x2 , x = 4.
10. Evaluate
R
(a) −2 +
(b) 2 −
1
sin 4
2
(c) −8 +
(d)
1
sin 4
2
1
sin 16
2
1
(sin 4 − 4 cos 4)
2
(e) * 8 −
1
sin 16
2
11. Find the volume under the surface z = 4x and above the triangle with vertices (0, 0), (1, 2) and (0, 4).
14
3
(b) 8
16
(c)
3
(d) 16
8
(e) *
3
(a)
ZZZ
12. Consider
2xy dV , where E is the region bounded by the parabolic cylinders x = y 2 and y = x2 and the planes
E
z = 0 and z = 3x + y. If we choose dV = dzdydx, which of the following is the resulting double integral?
Z 1 Z √x
(6x2 y + 2x2 y 2 ) dydx
(a)
x2
0
Z 1 Z x2
(b)
√
0
Z 1Z 1
(c)
0
(6x2 y + 2x2 y 2 ) dydx
x
(6x2 y + 2xy 2 ) dydx
0
Z 1 Z x2
(d)
√
0
(6x2 y + 2xy 2 ) dydx
Z 1
x
Z √
0
x2
(e) *
x
(6x2 y + 2xy 2 ) dydx
5
Part II: Work out. Show all intermediate steps.
ZZ
13. Consider
(2x + 3y + 1) dA, where R is the triangular region in the xy-plane with vertices (1, 1), (2, 3) and (3, 1).
R
Note: The equation of the two sloped lines in the xy-plane are y = 2x − 1 and y = 7 − 2x.
a.) Set up but do not evaluate the corresponding double integral in the order dy dx.
b.) Set up but do not evaluate the corresponding double integral in the order dx dy.
6
14. Evaluate
Z 0 Z √1−x2 p
−1
1 − x2 − y 2 dy dx by converting to polar coordinates. Simplify your answer. Note: no credit
0
will be awarded by evaluating the integral in rectangular coordinates.
7
x−y
dA, where R is the trapezoidal region with vertices
R x+y
(4, 0), (6, 0), (0, 4) and (0, 6) into an integral over a region S in the uv-plane. Do not evaluate the integral.
ZZ
15. Use the transformation u = x−y, v = x+y to rewrite
8
Z 4 Z √x Z 12−3x
16. Consider
0
√
− x
f (x, y, z) dz dy dx
0
(i) Rewrite this integral as an iterated integral in the order dx dz dy
(ii) Rewrite this integral as an iterated integral in the order dy dx dz
9