Deck 7: Applications of the Integral

Question 1

Let A denote the area enclosed by the equation $y = x ^ { 2 } - 4 x + 3$ and the x-axis. Then A is

A) $\frac { 4 } { 9 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 2 } { 3 }$

Answer 1

$\frac { 4 } { 3 }$

Question 2

Let A denote the area enclosed by the equation $y = \sin x$ and the x-axis with $x \in [ 0 , \pi ]$ Then A is

A) $\frac { 1 } { 2 }$
B)1
C) $\frac { 3 } { 2 }$
D)2
E) $\frac { 5 } { 2 }$

Answer 2

2

Question 3

Let A denote the area enclosed by the equation $y = 4 - | x |$ and the x-axis. Then A is

A)8
B)10
C)12
D)14
E)16

Answer 3

16

Question 4

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 2 x$ Then A is

A) $\frac { 4 } { 9 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 2 } { 3 }$

Question 5

Let A denote the area enclosed by the equations $y = x ^ { 2 } + 1$ and $y = - 4 x - 2$ Then A is

A) $\frac { 4 } { 9 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 2 } { 3 }$

Question 6

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 8 - x ^ { 2 }$ Then A is

A) $\frac { 4 } { 3 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 16 } { 3 }$
D) $\frac { 32 } { 3 }$
E) $\frac { 64 } { 3 }$

Question 7

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 6 x - x ^ { 2 }$ Then A is

A)3
B)6
C)9
D)12
E)18

Question 8

Let A denote the area enclosed by the y-axis and the equations y = sin x and y = cos x for $0 \leq x \leq \frac { \pi } { 4 }$ Then A is

A) $\frac { \sqrt { 2 } } { 2 }$
B) $\sqrt { 2 } - 1$
C) $\sqrt { 2 }$
D)1
E) $\sqrt { 2 } + 1$

Question 9

Let A denote the area enclosed by the equations y = x³ and y = x. Then A is

A) $\frac { 1 } { 2 }$
B)1
C) $\frac { 3 } { 2 }$
D)2
E) $\frac { 5 } { 2 }$

Question 10

Let A denote the area enclosed by the equations $y = 2 x ^ { 3 } - 3 x ^ { 2 } - 5 x$ and $y = x ^ { 3 } - 2 x ^ { 2 } - 3 x$ Then A is

A)2
B) $\frac { 27 } { 12 }$
C)3
D) $\frac { 37 } { 12 }$
E) $\frac { 39 } { 12 }$

Question 11

Let A denote the area enclosed by the equations $y = 3 - 2 | x |$ and $y = | x | \text {. }$ Then A is

A)2
B) $\frac { 9 } { 4 }$
C) $\frac { 11 } { 4 }$
D)3
E) $\frac { 13 } { 4 }$

Question 12

Let A denote the area enclosed by the equations $y = \frac { 3 } { x }$ and $y = 4 - x$ Then A is

A)2
B) $2 + \ln 3$
C)3
D) $4 - 3 \ln 3$
E) $3 + \ln 3$

Question 13

Let A denote the area enclosed by the equations $y = 2 | x |$ and $y = 3 - x$ Then A is

A)15
B)12
C)9
D)6
E)3

Question 14

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 2 - | x |$ Then A is

A) $\frac { 11 } { 3 }$
B)3
C) $\frac { 7 } { 3 }$
D) $\frac { 5 } { 3 }$
E) $\frac { 4 } { 3 }$

Question 15

Let A denote the area enclosed by the equations $y = \sqrt [ 3 ] { x }$ and $y = 4 x$ Then A is

A) $\frac { 1 } { 64 }$
B) $\frac { 1 } { 32 }$
C) $\frac { 1 } { 16 }$
D) $\frac { 1 } { 8 }$
E) $\frac { 1 } { 4 }$

Question 16

Let A denote the area enclosed by the equations $y ^ { 2 } = x - 2$ and $x = 4$ Then A is

A) $\frac { 64 } { 3 }$
B) $\frac { 32 } { 3 }$
C) $\frac { 16 } { 3 }$
D) $\frac { 8 \sqrt { 2 } } { 3 }$
E) $\frac { 4 } { 3 }$

Question 17

Let A denote the area enclosed by the equations $y ^ { 2 } = x$ and $y = 2 - x$ Then A is

A) $\frac { 11 } { 2 }$
B) $\frac { 9 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 3 } { 2 }$

Question 18

Let A denote the area enclosed by the equations $x = y ^ { 3 } - 2 y$ and $x = 7 y .$ Then A is

A) $\frac { 81 } { 2 }$
B) $\frac { 63 } { 2 }$
C) $\frac { 45 } { 2 }$
D) $\frac { 27 } { 2 }$
E) $\frac { 9 } { 2 }$

Question 19

Let A denote the area enclosed by the equations $x = y ^ { 2 }$ and $x = y + 6$ Then A is

A) $\frac { 625 } { 6 }$
B) $\frac { 225 } { 6 }$
C) $\frac { 125 } { 6 }$
D) $\frac { 25 } { 6 }$
E) $\frac { 5 } { 6 }$

Question 20

Let A denote the area enclosed by the equations $x = 3 y - y ^ { 2 }$ and x = y. Then A is

A) $\frac { 40 } { 3 }$
B) $\frac { 35 } { 3 }$
C) $\frac { 25 } { 3 }$
D)10
E) $\frac { 4 } { 3 }$

Question 21

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 x ^ { 2 },$ the x-axis, $x = 1 ;$ about the x-axis. Then V is

A) $\frac { 4 \pi } { 3 }$
B) $\frac { 4 \pi } { 5 }$
C) $\frac { 8 \pi } { 3 }$
D) $\frac { 8 \pi } { 5 }$
E) $\frac { \pi } { 5 }$

Question 22

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 | x | \text {, }$ the x-axis, $x = - 1 ;$ about the x-axis. Then V is

A) $\frac { 13 \pi } { 3 }$
B) $\frac { 11 \pi } { 3 }$
C) $\frac { 10 \pi } { 3 }$
D) $\frac { 4 \pi } { 3 }$
E) $\frac { 7 \pi } { 3 }$

Question 23

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 4;$ about the x-axis. Then V is

A) $8 \pi$
B) $7 \pi$
C) $6 \pi$
D) $5 \pi$
E) $4 \pi$

Question 24

Let V be the volume of the solid generated by revolving the region enclosed by $y = 3 x ^ { 2 },$ the x-axis, $x = 1 ;$ about the x-axis. Then V is

A) $\frac { 12 \pi } { 5 }$
B) $\frac { 11 \pi } { 5 }$
C) $\frac { 9 \pi } { 5 }$
D) $\frac { 8 \pi } { 5 }$
E) $\frac { 7 \pi } { 5 }$

Question 25

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, $y = 2;$ about the y-axis. Then V is

A) $\frac { 52 \pi } { 5 }$
B) $\frac { 42 \pi } { 5 }$
C) $\frac { 32 \pi } { 5 }$
D) $\frac { 22 \pi } { 5 }$
E) $\frac { 12 \pi } { 5 }$

Question 26

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 1;$ about the y-axis. Then V is

A) $\frac { 4 \pi } { 5 }$
B) $\frac { 8 \pi } { 5 }$
C) $\frac { 16 \pi } { 5 }$
D) $\frac { 32 \pi } { 5 }$
E) $\frac { 64 \pi } { 5 }$

Question 27

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { \sin x },$ the y-axis, about the x-axis. Then V is

A) $\pi$
B) $\frac { 3 \pi } { 2 }$
C) $2 \pi$
D) $\frac { 5 \pi } { 2 }$
E) $3 \pi$

Question 28

Let V be the volume of the solid generated by revolving the region bounded above by $y = \sqrt { \cos x }$ and bounded below by the x-axis from x = 0 to $x = \frac { \pi } { 2 };$ about the x-axis. Then V is

A) $\pi$
B) $\frac { 3 \pi } { 2 }$
C) $2 \pi$
D) $\frac { 5 \pi } { 2 }$
E) $3 \pi$

Question 29

Let V be the volume of the solid generated by revolving the region enclosed by $y = x ^ { 2 },$ the x-axis, $x = 2;$ about $x = 2$ Then V is

A) $\frac { 10 \pi } { 3 }$
B) $\frac { 11 \pi } { 5 }$
C) $\frac { 9 \pi } { 5 }$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 8 \pi } { 5 }$

Question 30

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, y = 4; about y = 4. Then V is

A) $\pi$
B) $\frac { 128 \pi } { 3 }$
C) $\frac { 128 \pi } { 5 }$
D) $\frac { 128 \pi } { 3 }$ .
E) $\frac { 64 \pi } { 5 }$

Question 31

Let V be the volume of the solid generated by revolving the region bounded above by $y = 2 x ^ { 2 }$ and bounded below the x-axis, $x \in [ 0 , \sqrt { 2 } ];$ about the y-axis. Then V is

A) $4 \pi$
B) $\frac { 128 \pi } { 3 }$
C) $\frac { 32 \pi } { 5 }$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 8 \pi } { 5 }$

Question 32

Let V be the volume of the solid generated by revolving the region enclosed by y = sec x, the x-axis, $x = \frac { \pi } { 3 };$ about the x-axis. Then V is

A) $\sqrt { 3 } \pi$
B) $\frac { 128 \pi } { 3 }$
C) $\frac { 32 \pi } { 5 }$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 8 \pi } { 5 }$

Question 33

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 4;$ about the y-axis. Then V is

A) $\sqrt { 3 } \pi$
B) $\frac { 128 \pi } { 5 }$
C) $\frac { 88 \pi } { 5 }$
D) $\frac { 8 \pi } { 5 }$
E) $\frac { 8 \pi } { 3 }$

Question 34

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 x ^ { 2 },$ the x-axis, $x = 1;$ about the y-axis. Then V is

A) $\sqrt { 3 } \pi$
B) $\frac { 128 \pi } { 3 }$
C) $\frac { 88 \pi } { 3 }$
D) $\frac { 8 \pi } { 3 }$
E) $\pi$

Question 35

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, $y = 2;$ about the x-axis. Then V is

A) $\sqrt { 3 } \pi$
B) $8 \pi$
C) $\frac { 88 \pi } { 3 }$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 8 \pi } { 5 }$

Question 36

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, x = 4; about $y = 3$ . Then V is

A) $\sqrt { 3 } \pi$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 8 \pi } { 5 }$

Question 37

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

A) $\sqrt { 3 } \pi$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 1408 \pi } { 15 }$
E) $\frac { 8 \pi } { 15 }$

Question 38

Let V be the volume of the solid generated by revolving the region enclosed by $x = y ^ { 2 } + 1 , x = 2 y + 1;$ about $x = - 1$ Then V is

A) $\frac { 48 \pi } { 5 }$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 1408 \pi } { 5 }$
E) $\frac { 8 \pi } { 5 }$

Question 39

Let V be the volume of the solid generated by revolving the region enclosed by about x = 1. Then V is

A) $\frac { 48 \pi } { 5 }$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 40

Let V be the volume of the solid generated by revolving the region enclosed by about x = 2. Then V is

A) $\frac { 48 \pi } { 5 }$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 41

Let V be the volume of the solid generated by revolving the region enclosed by y = x², y = x - x²; about the x-axis. Then V is

A) $\frac { \pi } { 96 }$
B) $8 \pi$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 42

Let V be the volume of the solid generated by revolving the region enclosed by x = y², x = 2y - y²; about the x-axis. Then V is

A) $\frac { \pi } { 48 }$
B) $8 \pi$
C) $\frac { \pi } { 3 }$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 43

Let V be the volume of the solid generated by revolving the region enclosed by y = x², y = x + 2; about the x-axis. Then V is

A) $\frac { \pi } { 48 }$
B) $\frac { 64 \pi } { 5 }$
C) $24 \pi$
D) $\frac { 72 \pi } { 5 }$
E) $\frac { 8 \pi } { 5 }$

Question 44

Let V be the volume of the solid generated by revolving the region enclosed by x = y², x = y + 2; about the y-axis. Then V is

A) $\frac { \pi } { 48 }$
B) $8 \pi$
C) $\frac { \pi } { 3 }$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 72 \pi } { 5 }$

Question 45

Let V be the volume of the solid generated by revolving the region enclosed by y = x³, y = 2x² - x³; about the y-axis. Then V is

A) $\frac { \pi } { 48 }$
B) $\frac { 64 \pi } { 5 }$
C) $\frac { \pi } { 5 }$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 46

Let V be the volume of the solid generated by revolving the region enclosed by x = y³, x = 2y² - y³; about the x-axis. Then V is

A) $\frac { \pi } { 48 }$
B) $8 \pi$
C) $\frac { \pi } { 3 }$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { \pi } { 5 }$

Question 47

Let V be the volume of the solid generated by revolving the region enclosed by y = -x², y = x² - 2x; about the y-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 48

Let V be the volume of the solid generated by revolving the region enclosed by x = -y², x = y² - 2y; about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 49

Let V be the volume of the solid generated by revolving the region enclosed by y = x², y = 2 - x; about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $\frac { 72 \pi } { 5 }$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 50

Let V be the volume of the solid generated by revolving the region enclosed by x = -y², x = y - 2; about the y-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 72 \pi } { 5 }$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 51

Let V be the volume of the solid generated by revolving the region enclosed by y = -x³, y = x³ - 2x²; about the y-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $\frac { 72 \pi } { 5 }$
D) $\frac { \pi } { 5 }$
E) $\frac { 8 \pi } { 5 }$

Question 52

Let V be the volume of the solid generated by revolving the region enclosed by x = -y³, x = y³ - 2y²; about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { \pi } { 5 }$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 53

Let V be the volume of the solid generated by revolving the region enclosed by y = 5 - x², y = 4; about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $\frac { 176 \pi } { 15 }$
D) $\frac { \pi } { 5 }$
E) $\frac { 8 \pi } { 5 }$

Question 54

Let V be the volume of the solid generated by revolving the region enclosed byabout the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 64 \pi } { 5 }$
C) $\frac { 176 \pi } { 15 }$
D) $\frac { \pi } { 5 }$
E) $\pi$

Question 55

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 704 \pi } { 5 }$
C) $\frac { 176 \pi } { 15 }$
D) $\frac { \pi } { 5 }$
E) $\pi$

Question 56

Let V be the volume of the solid generated by revolving the region enclosed by about the y-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 704 \pi } { 5 }$
C) $\frac { 176 \pi } { 15 }$
D) $\frac { 8 \pi } { 3 }$
E) $\pi$

Question 57

Let V be the volume of the solid generated by revolving the region enclosed by about the y-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 704 \pi } { 5 }$
C) $\frac { 2 \pi \left( 1 - \pi ^ { 2 } \right) } { 12 }$
D) $\frac { 8 \pi } { 3 }$
E) $\pi$

Question 58

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 704 \pi } { 5 }$
C) $\frac { \pi ( 4 - \pi ) } { 2 }$
D) $\frac { 8 \pi } { 3 }$
E) $\pi$

Question 59

Let V be the volume of the solid generated by revolving the region enclosed by about $x = - y ^ { 2 },$ Then V is

A) $\frac { 5 \pi } { 3 }$
B) $\frac { \pi } { 5 }$
C) $24 \pi$
D) $\frac { 1088 \pi } { 15 }$
E) $\frac { 8 \pi } { 5 }$

Question 60

Let V be the volume of the solid generated by revolving the region enclosed by about $y = - x ^ { 2 },$ Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 704 \pi } { 5 }$
C) $\frac { 176 \pi } { 15 }$
D) $\frac { 7 \pi } { 3 }$
E) $\pi$

Question 61

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -1 to x = 1. The cross-sections of this solid perpendicular to the x-axis run from $y = - 4 \sqrt { 1 - x ^ { 2 } }$ to $y = 4 \sqrt { 1 - x ^ { 2 } }$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 25 } { 3 }$
B) $\frac { 274 } { 5 }$
C) $\frac { 256 } { 3 }$
D) $\frac { 7 } { 3 }$
E)42

Question 62

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -3 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 25 } { 3 }$
B)144
C) $\frac { 256 } { 3 }$
D) $72$
E) $55$

Question 63

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 25 } { 3 }$
B)144
C) $\frac { 256 } { 3 }$
D) $\frac { 72 } { 35 }$
E)55

Question 64

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 9 }$ to $y = \sqrt { \frac { x } { 3 } }$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 27 } { 70 }$
B)144
C) $\frac { 256 } { 3 }$
D) $\frac { 72 } { 35 }$
E) $\frac { 25 } { 3 }$

Question 65

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² and they are squares with bases in the xy-plane. Then V is

A) $\frac { 27 } { 70 }$
B) $\frac { 25 } { 3 }$
C) $\frac { 256 } { 3 }$
D) $\frac { 72 } { 35 }$
E) $\frac { 32 } { 5 }$

Question 66

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 2187 } { 70 }$
B)144
C) $\frac { 256 } { 3 }$
D) $\frac { 72 } { 35 }$
E) $\frac { 32 } { 5 }$

Question 67

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = 3 - x$ and they are squares with bases in the xy-plane. Then V is

A) $\frac { 72 } { 35 }$
B)14
C) $\frac { 56 } { 3 }$
D)9
E) $\frac { 32 } { 5 }$

Question 68

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -3 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ and they are semicircles with diameters in the xy-plane. Then V is

A) $\frac { \pi } { 3 }$
B) $14 \pi$
C) $18 \pi$
D) $\frac { 7 \pi } { 3 }$
E) $\pi$

Question 69

Let V be the volume of a wedge cut from a right-circular cylinder with a radius of 1 inch by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of $\frac { \pi } { 3 }$ radian. Then V is

A) $\frac { 1 } { \sqrt { 3 } } \pi$
B) $\frac { 2 } { \sqrt { 3 } } \pi$
C) $\sqrt { 3 } \pi$
D) $\frac { 4 } { \sqrt { 3 } } \pi$
E) $\frac { 5 } { \sqrt { 3 } } \pi$

Question 70

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are semicircles with diameters in the xy-plane. Then V is

A) $\frac { \pi } { 3 }$
B) $\frac { 47 \pi } { 3 }$
C) $\frac { 25 \pi } { 3 }$
D) $\frac { 7 \pi } { 3 }$
E) $\frac { 9 \pi } { 35 }$

Question 71

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 9 }$ to $y = \sqrt { \frac { x } { 3 } }$ and they are semicircles with diameters in the xy-plane. Then V is

A) $\frac { 27 \pi } { 560 }$
B) $\frac { 47 \pi } { 3 }$
C) $\frac { 25 \pi } { 3 }$
D) $\frac { 7 \pi } { 3 }$
E) $\frac { 9 \pi } { 35 }$

Question 72

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² and they are semicircles with bases in the xy-plane. Then V is

A) $\frac { 47 \pi } { 3 }$
B) $\frac { 25 \pi } { 3 }$
C) $\frac { 7 \pi } { 3 }$
D) $\frac { 9 \pi } { 35 }$
E) $\frac { 4 \pi } { 5 }$

Question 73

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are semicircles with diameters in the xy-plane. Then V is

A) $\frac { 2187 \pi } { 560 }$
B) $\frac { 25 \pi } { 3 }$
C) $\frac { 7 \pi } { 3 }$
D) $\frac { 9 \pi } { 35 }$
E) $\frac { 4 \pi } { 5 }$

Question 74

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² and they are semicircles with bases in the xy-plane. Then V is

A) $\frac { 2187 \pi } { 560 }$
B) $\frac { 25 \pi } { 3 }$
C) $\frac { 7 \pi } { 3 }$
D) $\frac { 9 \pi } { 35 }$
E) $\frac { 4 \pi } { 5 }$

Question 75

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = 3 - x$ and they are semicircles with diameters in the xy-plane. Then V is

A) $\frac { 25 \pi } { 3 }$
B) $\frac { 7 \pi } { 3 }$
C) $\frac { 9 \pi } { 35 }$
D) $\frac { 9 \pi } { 8 }$
E) $\frac { 4 \pi } { 5 }$

Question 76

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -3 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

A) $18 \sqrt { 2 }$
B) $36 \sqrt { 3 }$
C)36
D) $18 \sqrt { 5 }$
E)9

Question 77

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -1 to x = 1. The cross-sections of this solid perpendicular to the x-axis run from $y = - 4 \sqrt { 1 - x ^ { 2 } }$ to $y = 4 \sqrt { 1 - x ^ { 2 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

A) $\frac { 256 } { \sqrt { 3 } }$
B) $\frac { 128 } { \sqrt { 3 } }$
C) $\frac { 64 } { \sqrt { 3 } }$
D) $\frac { 32 } { \sqrt { 3 } }$
E) $\frac { 16 } { \sqrt { 3 } }$

Question 78

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are equilateral triangles with bases in the xy-plane. Then V is

A) $\frac { 72 \sqrt { 3 } } { 35 }$
B) $\frac { 36 \sqrt { 3 } } { 33 }$
C) $\frac { 18 \sqrt { 3 } } { 33 }$
D) $\frac { 18 \sqrt { 3 } } { 35 }$
E) $\frac { 12 \sqrt { 3 } } { 35 }$

Question 79

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 9 }$ to $y = \sqrt { \frac { x } { 3 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

A) $\frac { 27 \sqrt { 3 } } { 280 }$
B) $\frac { 27 \sqrt { 3 } } { 140 }$
C) $\frac { 27 \sqrt { 3 } } { 70 }$
D) $\frac { 27 \sqrt { 3 } } { 35 }$
E) $\frac { 27 \sqrt { 3 } } { 350 }$

Question 80

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = x ^ { 2 }$ and they are equilateral triangles with bases in the xy-plane. Then V is

A) $\frac { 64 \sqrt { 3 } } { 5 }$
B) $\frac { 56 \sqrt { 3 } } { 5 }$
C) $\frac { 48 \sqrt { 3 } } { 5 }$
D) $\frac { 32 \sqrt { 3 } } { 5 }$
E) $\frac { 8 \sqrt { 3 } } { 5 }$