Math 6C – Chapter 13 – Take-Home Quiz
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1. Let D denote the rectangle 1 < x < 3, 0 < y < 1. Then
(a)
9
2  y2
D
e e e e
4
12
x
 xy e
e12  e9  e2  e
4
2
e10  e9  e3  e
(c)
4
dxdy  ?
(b)
e10  e9  e2  e
(d)
4
e10  e8  e2  e
(e)
4
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R sin  x  cos  y  dxdy  ?
2. If R denotes the region described by 0 < x < /2, y < 2x, y > x, then
(a)
8  3
12
4 
3
(b)
(c)
3  8
12
3  2
6
(d)
(e)
4  3
12
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3. Let B be the rectangular solid defined by 0 < x < /2, 0 < y < 1, 0 < z < .
xyz sin  x

B
2
 cos  z  dx dy dz  ?
2
 
cos 
2
(a)
(c)



  2 
1  sin 

 4  




2

 
 
(b)
16
(d)

 2 
 1  cos
4


sin 

 2 
 1  cos
 4 
sin 
8

cos  2 1  cos 

 2

 
Then,
 2 
 
2
8

 
(e)
16


sin  2 1  cos 

 4


2

 
 
8
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4. Let W be the region 0 < x < 1, x < y < 2x, xy < z < 2xy, then
(a) 23/20
(b) 5/4
(c) 27/20
 x  y  z  dx dy dz  ?

W
(d) 29/20
(e) 31/20
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5. If S denotes the unit-hemisphere above the xy-plane, then
(a) /4
(b) /5
(c) 2/3
x

S
2
 y 2  z 2  dxdy dz  ?
(d) 2/5
(e) 3/4
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6. If R denotes the region y > 2(x–1), y < 2x + 1, y >1–x, y < 2–x, then
R xy dxdy  ?
( Hint: Change coordinates to u = y – 2x, v = x + y )
(a) 51/107
(b) 51/109
(c) 53/108
(d) 53/107
(e) 51/107
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7. Let R denote the region y > x, y < x + 1, y >1/x, y < 2/x. Then
(Hint: Change coordinates to u = y – x, v = xy )
(a) 1
(b) 3/2
(c) 2
R
 y  x
2
 4 xy dxdy  ?
(d) 5/2
(e) 3
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1
(a) 2 /3
(b) 3 /4
1 x2
  1 x
1
8. Change from Cartesian to polar coordinates to compute
2
1 x
2
2
y
2

2
dy dx .
(d) 
(c) 4 /5
(e) 3 /2
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9. Evaluate
 xyz dxdy dz
over the solid ellipsoid
x2 y 2 z 2
  1 .
a 2 b2 c 2
( Hint: let x = au, y = bv, z = cw, then integrate over an appropriate region in uvw–space )
(a)
 abc 
2
3
(b)
 abc 
 abc 
2
(c)
4
2
(d)
5
 abc 
2
(e) none of these
6
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3
10. Let D be the unit circle in the xy–plane. Then
(a)  /3
2
2
 ( x  y ) dxdy  ?
D
(b)  /4
(c)  /5
(d) 2
(e) 3 /4
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 x  u3  v2

11. Let 
define a coordinate transformation in some region of the plane.
2
4
 y  u  v
Then
dxdy 
?
(a) 6uv  2u 2v3
(c) 4uv  6u3v2
dudv.
(b) 12uv  4u3v2 
(d) 2uv  6u 2v3
(e) 4uv 12u 2v3
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12. Find the volume of the finite region enclosed by the two paraboloids,
f ( x, y)  x2  y 2 1, g ( x, y)  1  x 2  y 2 .
(a) 2 /3
(b) 3 /4
(d) 
(c) 4 /5
(e) 3 /2
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 x  euv
13. Let 
u v define a coordinate transformation in some region of the plane.
 y  e
Then
dxdy 
?
(a) (v  u)euvuv
(c) (v  u)euvuv
dudv.
(b) (v  u)euvuv 
(d) (v  u)euvuv
(e) (v  u )eu vuv
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KEY – 6C Chapter 13 Quiz
1– D
2– A
3– E
4– C
5– D
6– C
7– A
8– D
9– D
10– B
11– E
12– D
13– C