MAS117 Homework 4
Due: See Google Classroom
20
23
Instructions: Answer all of the following questions. Marks will not be given for answers
without showing the steps taken to arrive at the answer in a clear and logical manner. Clearly
label the question number and the final answer.
2/
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1. Consider the solid region S that lies under the surface z = x2 y and above the rectangle
R = [0, 2] ⇥ [1, 4].
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4
(a) Find a formula for the area of a cross-section of S in the plane perpendicular to the x-axis
at x for 0  x  2. Then use the formula to compute the areas of the cross-sections
56
illustrated. Answer: k = 1, A = 14
3 and k = 2, A = 3
M
AS
11
7
(b) Find a formula for the area of a cross-section of S in the plane perpendicular to the y-axis
at y for 1  y  4. Then use the formula to compute
the areas of the cross-sections
p
8 3
8
illustrated. Answer: k = 1, A = 3 and k = 3, A = 3
(c) Find the volume of S. Answer: 112
9
2. Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y 2
above the rectangle R = [ 1, 1] ⇥ [1, 2]. Answer: 52
3
x2 + 2 and
3. Find the volume of the solid enclosed by the surface z = x2 + xy 2 and the planes z = 0, x =
0, x = 5 and y = ±2. Answer: 700
3
4. Find the volume of the given solid.
(a) Enclosed by the paraboloid z = x2 + y 2 + 1 and the planes x = 0, y = 0, z = 0, and
x + y = 2. Answer: 14
3
(b) Bounded by the planes z = x, y = x, x + y = 2, and z = 0. Answer: 13
5. Use polar coordinates to find the volume of the given solid.
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(a) Below the cone z = x2 + y 2 and above the ring 1  x2 + y 2  4. Answer: 14⇡
3
p
(b) Inside the sphere x2 + y 2 + z 2 = 16 and outside the cylinder x2 + y 2 = 4. Answer: 32 3⇡
1
6. Find the area of the surface.
2
2
(a) The
p part of the plane 6x + 4y + 2z = 1 that lies inside the cylinder x + y = 25. Answer:
25 14⇡
7. Use a triple integral to find the volume of the given solid.
x2
z 2 . Answer: 16⇡
2/
(a) The solid enclosed by the paraboloids y = x2 + z 2 and y = 8
20
23
(b) The part of the sphere x2 + y 2 + z 2 = 4z that lies inside the paraboloid z = x2 + y 2 .
Answer: 4⇡
1 and y + z = 4.
4
(b) The solid enclosed by the cylinder x2 + z 2 = 4 and the planes y =
Answer: 20⇡
ew
or
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8. The average value of a function f (x, y, z) over a solid region E is defined to be
ZZZ
1
favg =
f (x, y, z) dV
V (E)
E
where V (E) is the volume of E. For instance, if ⇢ is a density function, then ⇢avg is the average
density of E.
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Find the average height of the points in the solid hemisphere x2 + y 2 + z 2  1, z
9. Given f (x, y, z) = xy.
0. Answer:
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3
8
(a) Express the triple integral
RRR
f (x, y, z) dV as an iterated integral in cylindrical coordinates
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11
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for the region E labeled in the figure below.
(b) Evaluate the iterated integral. Answer: 0
10. Find the volume of the smaller wedge cut from a sphere of radius a by two planes that
intersect along a diameter at an angle of ⇡/6. Answer: 19 ⇡a3
11. Evaluate the integral by changing to spherical coordinates.
Z Z p
Z p
a
a
a2 y 2
p
a2 y 2
a 2 x2 y 2
p
(x2 z + y 2 z + z 3 ) dz dx dy
a 2 x2 y 2
Answer: 0
12. Evaluate the line integral, where C is the given space curve.
p
R
(a) C y 2 z ds, C is the line segment from (3, 1, 2) to (1, 2, 5). Answer: 107
14
12
2
(b)
R
C (y dx + z dy + x dz), C : x =
13. Evaluate the line integral
R
p
t, y = t, z = t2 , 1  t  4. Answer: 722
15
C F · dr, where C is given by the vector function r(t).
23
(a) F (x, y, z) = (x + y 2 )i + xzj + (y + z)k, r(t) = t2 i + t3 j 2tk, 0  t  2. Answer: 8
(b) F (x, y, z) = xzi + z 3 j + yk, r(t) = et i + e2t j + e t k, 1  t  1. Answer: 2(e 1 e)
20
14. Find the work done by the force field F (x, y) = x2 i + yex j on a particle that moves along
the parabola x = y 2 + 1 from (1, 0) to (2, 1). Answer: 12 e2 12 e + 73
k
ew
or
y 2 = 4 and x2 + y 2 = 9. Answer: 195
2 ⇡
R
16. Use Green’s Theorem to evaluate C F · dr.
4
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15. Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.
R
(a) C y 4 dx + 2xy 3 dy , C is the ellipse x2 + 2y 2 = 2. Answer: 0
⌘
R ⇣
2
(b) C (1 y 3 ) dx + (x3 + ey ) dy , C is the boundary of the region between the circles x2 +
m
(a) F (x, y) = he x + y 2 , e y + x2 i, C consists of the arc of the curve y = cos x from ( ⇡/2, 0)
to (⇡/2, 0) and the line segment from (⇡/2, 0) to ( ⇡/2, 0). Answer: 12 ⇡
p
(b) F (x, y) = h x2 + 1, arctan xi, C is the triangle from (0, 0) to (1, 1) to (0, 1) to (0, 0).
Answer: ⇡4 12 ln 2
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17. Determine whether or not the vector field is conservative. If it is conservative, find a
function f such that F = rf .
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11
7
(a) F (x, y, z) = hyz, xz + y, xy xi
(b) F (x, y, z) = yz sin xyi + xz sin xyj cos xyk
(c) F (x, y, z) = ez cos xi + ey cos zj + (ez sin x ey sin z)k
RR
18. Evaluate the surface integral S F ·dS for the given vector field F and the oriented surface
S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
orientation.
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(a) F (x, y, z) = yzi + zxj + xyk, S is the surface z = x sin y, 0  x  2, 0  y  ⇡, with
upward orientation. Answer: 12 ⇡ 2
(b) F (x, y, z) = xi+yj+5k, S is the boundary of the region enclosed by the cylinder x2 +z 2 = 1
and the planes y = 0 and x + y = 2. Answer: 4⇡
R
19. Use Stokes’ Theorem to evaluate C F · dr. In each case C is oriented counterclockwise as
viewed from above, unless otherwise stated.
(a) F (x, y, z) = zex i + (z y 3 )j + (x z 3 )k, C is the circle y 2 + z 2 = 4, x = 3, oriented
clockwise as viewed from the origin. Answer: 4⇡
(b) F (x, y, z) = hx3 z, xy, y + z 2 i, C is the curve of intersection of the paraboloid z = x2 + y 2
and the plane z = x. Answer: ⇡4
RR
20. Use the Divergence Theorem to calculate the surface integral S F · dS.
(a) F (x, y, z) = (x3 + y 3 )i + (y 3 + z 3 )j + (z 3 + x3 )k, S is the sphere with center the origin and
radius 2. Answer: 384
5 ⇡
p
2
(b) F (x, y, z) = (xy z )i + x3 zj + (xy + z 2 )k, S is the surface of the solid bounded by the
32
cylinder x = y 2 and the planes x + z = 1 and z = 0. Answer: 105
21. Let F (x, y, z) = z arctan y 2 i + z 3 ln x2 + 1 j + zk. Find the flux of F across the part of
the paraboloid x2 + y 2 + z = 2 that lies above the plane z = 1 and is oriented upward. Answer:
3⇡
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