Deck 15: Multiple Integrals

Question 1

The partial integral $\int _ { 1 } ^ { 2 } \frac { x } { y ^ { 2 } } d y$ is

A) $\frac { x } { 42 }$
B) $\frac { x } { 3 }$
C) $\frac { x } { 2 }$
D) $2 x$
E) $3 x$

Answer 1

$\frac { x } { 2 }$

Question 2

The partial integral $\int _ { 0 } ^ { 2 } \frac { 2 x } { y } d x$ is

A) $\frac { 3 } { y }$
B) $\frac { 4 } { y }$
C) $\frac { 6 } { y }$
D) $\frac { 3 } { y ^ { 2 } }$
E) $\frac { 3 } { 2 y ^ { 2 } }$

Answer 2

$\frac { 4 } { y }$

Question 3

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } 2 x \cos ( y ) d x \right] d y$ is

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

Answer 3

$\frac { 1 } { \sqrt { 2 } }$

Question 4

The iterated integral $\int _ { 0 } ^ { 1 } \left[ \int _ { 0 } ^ { 3 } 3 x ^ { 2 } y ^ { 2 } d y \right] d x$ is

A)9
B)6
C)4
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 4 }$

Question 5

The iterated integral $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { \pi } 2 x \sin ( y ) d x \right] d y$ is

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

Question 6

The iterated integral $\int _ { 1 } ^ { e } \left[ \int _ { 0 } ^ { 2 } \frac { x } { y } d x \right] d y$ is

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

Question 7

The iterated integral $\int _ { 0 } ^ { 1 } \left[ \int _ { 0 } ^ { 1 } e ^ { x } d x \right] d y$ is

A) $e + 2$
B) $e + 1$
C) $e - 1$
D) $e - 2$
E) $e$

Question 8

By Fubini's Theorem, the double integral $\iint$ R $\left( 2 x + 3 y ^ { 2 } \right) d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 2,0 \leq y \leq 3 \}$ is

A)64
B)66
C)72
D)76
E)84

Question 9

By Fubini's Theorem, the double integral $\iint$ R $6 \left( x ^ { 2 } - y \right) d A$ with $R = \{ ( x , y ) : - 1 \leq x \leq 2,0 \leq y \leq 1 \}$ is

A)2
B)4
C)6
D)9
E)12

Question 10

By Fubini's Theorem, the double integral $\iint$ R $\sqrt { x y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

A) $\frac { 41 } { 9 }$
B) $\frac { 29 } { 4 }$
C) $\frac { 13 } { 9 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 4 } { 9 }$

Question 11

By Fubini's Theorem, the double integral $\iint$ R $4 x e ^ { y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

A) $e - 1$
B) $e + 1$
C) $2 ( e - 1 )$
D) $2 ( e + 1 )$
E) $e + 2$

Question 12

By Fubini's Theorem, the double integral $\iint$ R $2 x \tan ( y ) d A$ with $R = \left\{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq \frac { \pi } { 4 } \right\}$ is

A) $\frac { \ln 3 } { 2 }$
B) $\frac { \ln 2 } { 2 }$
C) $\frac { \ln 2 } { 4 }$
D) $\frac { \ln 2 } { 3 }$
E) $\frac { \ln 3 } { 4 }$

Question 13

By Fubini's Theorem, the double integral $\iint$ R $\frac { 2 x } { y + 1 } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq e - 1 \}$ is

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

Question 14

By Fubini's Theorem, the double integral $\iint$ R $y e ^ { x } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 2 \}$ is

A) $e - 1$
B) $e + 1$
C) $2 ( e - 1 )$
D) $2 ( e + 1 )$
E) $e + 2$

Question 15

By Fubini's Theorem, the double integral $\iint$ R $\frac { y } { x } d A$ with $R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}$ is

A) $\ln 2$
B) $\frac { \ln 2 } { 2 }$
C) $\ln 3$
D) $\ln 4$
E) $\frac { \ln 3 } { 2 }$

Question 16

By Fubini's Theorem, the double integral $\iint$ R $\frac { y } { x ^ { 2 } } d A$ with $R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}$ is

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

Question 17

The volume of the surface bounded by the graph of $z = 4 x + 2 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

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

Question 18

The volume of the surface bounded by the graph of $z = 6 x + 4 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

A)2
B)3
C)4
D)5
E)7

Question 19

The volume of the surface bounded by the graph of $z = \frac { 1 } { x } + \frac { 1 } { y }$ with $1 \leq x \leq 2,1 \leq y \leq 2$ is

A) $\ln 2$
B) $\ln 4$
C) $\ln 3$
D) $\ln \frac { 1 } { 4 }$
E) $\frac { \ln 3 } { 2 }$

Question 20

The volume of the surface bounded by the graph of $z = e ^ { x + y }$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

A) $e - 1$
B) $e + 1$
C) $( e - 1 ) ^ { 2 }$
D) $( e + 1 ) ^ { 2 }$
E) $( e + 2 ) ^ { 2 }$

Question 21

The iterated integral $\int _ { 1 } ^ { 2 } \left[ \int _ { 0 } ^ { 2 x } x y ^ { 3 } d y \right] d x$ is

A)56
B)48
C)42
D)36
E)32

Question 22

The iterated integral $\int _ { 0 } ^ { 4 } \left[ \int _ { 0 } ^ { y } d x \right] d y$ is

A)8
B)12
C)14
D)16
E)18

Question 23

The iterated integral $\int _ { 0 } ^ { 4 } \left[ \int _ { 0 } ^ { y } \sqrt { 9 + y ^ { 2 } } d x \right] d y$ is

A) $\frac { 68 } { 3 }$
B) $\frac { 79 } { 3 }$
C) $\frac { 83 } { 3 }$
D) $\frac { 98 } { 3 }$
E) $\frac { 101 } { 3 }$

Question 24

The iterated integral $\int _ { - 1 } ^ { 1 } \left[ \int _ { 1 } ^ { e ^ { z } } \frac { x } { y } d y \right] d x$ is

A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 5 } { 3 }$
E) $\frac { 8 } { 3 }$

Question 25

The iterated integral $\int _ { 1 } ^ { 4 } \left[ \int _ { y ^ { 2 } } ^ { y } \sqrt { \frac { y } { x } } d x \right] d y$ is

A) $- \frac { 31 } { 5 }$
B) $- \frac { 33 } { 5 }$
C) $- \frac { 39 } { 5 }$
D) $- \frac { 41 } { 5 }$
E) $- \frac { 49 } { 5 }$

Question 26

The iterated integral $\int _ { 1 } ^ { 4 } \left[ \int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { y } { x } } d y \right] d x$ is

A) $- \frac { 473 } { 21 }$
B) $- \frac { 451 } { 21 }$
C) $- \frac { 403 } { 21 }$
D) $- \frac { 401 } { 21 }$
E) $- \frac { 383 } { 21 }$

Question 27

The iterated integral $\int _ { 0 } ^ { 3 } \left[ \int _ { 0 } ^ { x } x ^ { 2 } e ^ { x y } d y \right] d x$ is

A) $\frac { e ^ { 9 } + 10 } { 2 }$
B) $\frac { e ^ { 9 } - 10 } { 2 }$
C) $\frac { e ^ { 7 } + 10 } { 2 }$
D) $\frac { e ^ { 7 } - 10 } { 2 }$
E) $\frac { e ^ { 5 } + 10 } { 2 }$

Question 28

The iterated integral $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { x } \sin ( 4 x - y ) d y \right] d x$ is

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

Question 29

If R is the region bounded by $y = 2 x , y = \frac { x } { 2 }$ and $x = \frac { \pi } { 2 }$ then $\iint$ R $\sin x d A$ is

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

Question 30

If R is the region bounded by y = x, x = ? and the x-axis, then $\iint$ R $\cos ( x + y ) d A$ is

A) $- 3$
B) $- 2$
C) $- 1$
D) $- \frac { 1 } { 2 }$
E) $- \frac { 1 } { 4 }$

Question 31

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _{R $x ^ { 2 } \sqrt { 9 - x ^ { 2 } } d A$} is

A) $\frac { 584 } { 5 }$
B) $\frac { 632 } { 5 }$
C) $\frac { 694 } { 5 }$
D) $\frac { 744 } { 5 }$
E) $\frac { 648 } { 5 }$

Question 32

If R is the region bounded by y = x, y = 2, and xy = 1, then $\iint$ _R $\frac { y ^ { 2 } } { x ^ { 2 } } d A$ is

A) $\frac { 15 } { 4 }$
B) $\frac { 13 } { 4 }$
C) $\frac { 11 } { 4 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 5 } { 4 }$

Question 33

The area bounded by $y = x ^ { 3 }$ and $y = x ^ { 2 }$ is

A) $\frac { 13 } { 12 }$
B) $\frac { 11 } { 12 }$
C) $\frac { 7 } { 12 }$
D) $\frac { 5 } { 12 }$
E) $\frac { 1 } { 12 }$

Question 34

The area bounded by $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ is

A) $\frac { 17 } { 3 }$
B) $\frac { 16 } { 3 }$
C) $\frac { 14 } { 3 }$
D) $\frac { 13 } { 3 }$
E) $\frac { 11 } { 3 }$

Question 35

The area bounded by $y = x ^ { 2 } - 9$ and $y = 9 - x ^ { 2 }$ is

A)72
B)68
C)64
D)60
E)56

Question 36

The area bounded by $x ^ { 2 } + y ^ { 2 } = 16$ and $y ^ { 2 } = 6 x$ is

A) $\frac { 4 ( 9 \sqrt { 3 } + 2 + 6 \pi ) } { 9 }$
B) $\frac { 4 ( 9 \sqrt { 3 } + 6 \pi ) } { 9 }$
C) $\frac { 4 ( 9 \sqrt { 3 } - 2 + 6 \pi ) } { 9 }$
D) $\frac { - 4 ( 9 \sqrt { 3 } + 6 \pi ) } { 9 }$
E) $\frac { - 4 ( 9 \sqrt { 3 } + 2 + 6 \pi ) } { 9 }$

Question 37

If the order of integration of $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { \sqrt { x } } } ^ { 2 } f ( x , y ) d y \right] d x$ is switched, the result is

A) $\int _ { 0 } ^ { 4 } \left[ \int _ { 4 } ^ { y ^ { 2 } } f ( x , y ) d x \right] d y$
B) $\int _ { 0 } ^ { 2 } \left[ \int _ { 2 } ^ { y ^ { 2 } } f ( x , y ) d x \right] d y$
C) $\int _ { 0 } ^ { 2 } \left[ \int _ { 4 } ^ { y ^ { 2 } } f ( x , y ) d x \right] d y$
D) $\int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { y ^ { 2 } } f ( x , y ) d x \right] d y$
E) $\int _ { 0 } ^ { 4 } \left[ \int _ { 2 } ^ { y ^ { 2 } } f ( x , y ) d x \right] d y$

Question 38

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

A) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
B) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
D) $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
E) $\int _ { 0 } ^ { 1 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$

Question 39

If the order of integration of $\int _ { 0 } ^ { 1 } \left[ \int _ { y ^ { 2 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$ is switched, the result is

A) $\int _ { 1 } ^ { 3 } \left[ \int _ { x ^ { 3 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
B) $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { x ^ { 2 } } f ( x , y ) d y \right] d x$
C) $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 3 } } ^ { \sqrt { z } } f ( x , y ) d y \right] d x$
D) $\int _ { 0 } ^ { 1 } \left[ \int _ { \sqrt { x } } ^ { x ^ { 3 } } f ( x , y ) d y \right] d x$
E) $\int _ { 0 } ^ { 1 } \left[ \int _ { x ^ { 3 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$

Question 40

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

A) $\int _ { 0 } ^ { 8 } \left[ \int _ { \sqrt [ 4 ] { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
B) $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$
D) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
E) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$

Question 41

For what value of c is the function
$F ( x , y ) = c x y , 0 \leq x \leq 2,0 \leq y \leq 4$ 0, elsewhere
A joint probability density function for random variables X and Y?

A)1/16
B)1/8
C)1/4
D)1/2

Question 42

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 9 , x \geq 0 , y \geq 0$ then $\iint$ _R $( 2 x + 5 y ) d A$ in polar form is

A) $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } ( 2 \cos \theta + 5 \sin \theta ) d r \right] d \theta$
B) $\int _ { - \pi } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } ( 2 \cos \theta + 5 \sin \theta ) d r \right] d \theta$
C) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } ( 2 \cos \theta + 5 \sin \theta ) d r \right] d \theta$
D) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } ( 2 \cos \theta + 5 \sin \theta ) d r \right] d \theta$
E) $\int _ { 0 } ^ { 2 \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } ( 2 \cos \theta + 5 \sin \theta ) d r \right] d \theta$

Question 43

If R is the region bounded by $y = \sqrt { 4 - x ^ { 2 } } , y = x , y = - x$ then $\iint$ _{R $\left( x ^ { 2 } + y ^ { 2 } \right) d A$} in polar form is

A) $\int _ { \frac { \pi } { 2 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$
B) $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$
C) $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$
D) $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$
E) $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$

Question 44

If R is the region bounded by $x = \sqrt { 1 - y ^ { 2 } } , y = x$ and the positive x-axis, then $\iint$ _{R $x d A$} in polar form is

A) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$
B) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$
C) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$
E) $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$

Question 45

If R is the region bounded by $y = \sqrt { 9 - x ^ { 2 } }$ in the first quadrant, then $\iint$ _{R $y d A$} in polar form is

A) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$
B) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$
C) $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$
E) $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$

Question 46

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 4 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _R $\left( x ^ { 2 } + y \right) d A$ in polar form is

A) $\int _ { 0 } ^ { 2 \pi } \left[ \int _ { 2 } ^ { 3 } r \left( r \cos ^ { 2 } \theta + \sin \theta \right) d r \right] d \theta$
B) $\int _ { 0 } ^ { 2 \pi } \left[ \int _ { 2 } ^ { 3 } r ^ { 2 } \left( r \cos ^ { 2 } \theta + \sin \theta \right) d r \right] d \theta$
C) $\int _ { 0 } ^ { \pi } \left[ \int _ { 2 } ^ { 3 } r ^ { 2 } \left( r \cos ^ { 2 } \theta + \sin \theta \right) d r \right] d \theta$
D) $\int _ { 0 } ^ { \pi } \left[ \int _ { 2 } ^ { 3 } r \left( r \cos ^ { 2 } \theta + \sin \theta \right) d r \right] d \theta$
E) $\int _ { - \pi } ^ { \pi } \left[ \int _ { 2 } ^ { 3 } r ^ { 2 } \left( r \cos ^ { 2 } \theta + \sin \theta \right) d r \right] d \theta$

Question 47

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _R $5 \sqrt { x ^ { 2 } + y ^ { 2 } } d A$ in polar form is

A) $\int _ { 0 } ^ { \pi } \left[ \int _ { 1 } ^ { 3 } 5 r ^ { 2 } d r \right] d \theta$
B) $\int _ { 0 } ^ { 2 \pi } \left[ \int _ { 1 } ^ { 3 } 5 r d r \right] d \theta$
C) $\int _ { 0 } ^ { 2 \pi } \left[ \int _ { 1 } ^ { 3 } 5 r ^ { 2 } d r \right] d \theta$
D) $\int _ { 0 } ^ { \pi } \left[ \int _ { 1 } ^ { 3 } 5 r d r \right] d \theta$
E) $\int _ { \pi } ^ { 2 \pi } \left[ \int _ { 1 } ^ { 3 } 5 r ^ { 2 } d r \right] d \theta$

Question 48

If R is the region in the first quadrant bounded by $x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _R $e ^ { x ^ { 2 } + y ^ { 2 } } d A$ in polar form is

A) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } e ^ { r 2 } d r \right] d \theta$
B) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$
C) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 2 } ^ { 3 } e ^ { r 2 } d r \right] d \theta$
E) $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$

Question 49

Let $I = \int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } d y \right] d x$ Then I in polar form is

A) $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } r d r \right] d \theta$
B) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } d r \right] d \theta$
C) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } r d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } d r \right] d \theta$
E) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r d r \right] d \theta$

Question 50

Let $I = \int _ { - 3 } ^ { 3 } \left[ \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d y \right] d x$ Then I in polar form is

A) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } d r \right] d \theta$
B) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 3 } d r \right] d \theta$
C) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 3 } d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } d r \right] d \theta$
E) $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r ^ { 3 } d r \right] d \theta$

Question 51

Let $I = \int _ { 0 } ^ { 1 } \left[ \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } d y \right] d x$ Then I in polar form is

A) $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { 1 } r \sqrt { 1 - r ^ { 2 } } d r \right] d \theta$
B) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 1 } \sqrt { 1 - r ^ { 2 } } d r \right] d \theta$
C) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } \sqrt { 1 - r ^ { 2 } } d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r \sqrt { 1 - r ^ { 2 } } d r \right] d \theta$
E) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 1 } r \sqrt { 1 - r ^ { 2 } } d r \right] d \theta$

Question 52

Let $I = \int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \sin \left( x ^ { 2 } + y ^ { 2 } \right) d y \right] d x$ Then I in polar form is

A) $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } r \sin \left( r ^ { 2 } \right) d r \right] d \theta$
B) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } \sin \left( r ^ { 2 } \right) d r \right] d \theta$
C) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 2 } r \sin \left( r ^ { 2 } \right) d r \right] d \theta$
D) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r \sin \left( r ^ { 2 } \right) d r \right] d \theta$
E) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } \sin \left( r ^ { 2 } \right) d r \right] d \theta$

Question 53

Let $I = \int _ { 0 } ^ { 3 } \left[ \int _ { 0 } ^ { \sqrt { 9 - y ^ { 2 } } } e ^ { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x \right] d x$ Then I in polar form is

A) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } e ^ { r } d r \right] d \theta$
B) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r e ^ { r } d r \right] d \theta$
C) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r e ^ { r } d r \right] d \theta$
D) $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } e ^ { r } d r \right] d \theta$
E) $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { 3 } r e ^ { r } d r \right] d \theta$

Question 54

The area of the region inside $r = 2 ( 1 + \cos \theta )$ and outside r = 2 is

A) $8 + \frac { \pi } { 2 }$
B) $8 - \frac { \pi } { 2 }$
C) $8 + \pi$
D) $8 - \pi$
E) $\frac { 8 + \pi } { 2 }$

Question 55

The area of one leaf of rose $r = 2 \cos ( 2 \theta )$ is

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

Question 56

The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

A)9?
B)81?
C)18?
D)24?
E)36?

Question 57

The volume of the solid cut out of $z ^ { 2 } + r ^ { 2 } = 4$ by r = 1 is

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

Question 58

The volume of the solid cut out of $z ^ { 2 } + r ^ { 2 } = 16$ by $r = 4 \cos \theta$ is

A) $\frac { 128 ( 3 \pi - 4 ) } { 3 }$
B) $\frac { 128 ( 3 \pi - 4 ) } { 9 }$
C) $\frac { 128 ( 3 \pi + 4 ) } { 9 }$
D) $\frac { 128 ( 3 \pi - 2 ) } { 9 }$
E) $\frac { 128 ( 3 \pi + 2 ) } { 9 }$

Question 59

The volume of the solid above the polar plane bounded by z = 2r and $r = 1 - \cos \theta$ is

A) $\frac { 19 \pi } { 3 }$
B) $\frac { 17 \pi } { 3 }$
C) $\frac { 13 \pi } { 3 }$
D) $\frac { 11 \pi } { 3 }$
E) $\frac { 10 \pi } { 3 }$

Question 60

The volume of the solid bounded by $z = 4 - r ^ { 2 } , r = 1$ and the polar plane is

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

Question 61

The volume of the solid bounded above by $z = r ^ { 2 }$ and below by $z = 2 r \sin \theta$ is

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

Question 62

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, x = 3, y = 2 and the y-axis with area density $\rho ( x , y ) = x y ^ { 2 }$ is

A) $\left( - 1 , \frac { 3 } { 4 } \right)$
B) $\left( 1 , - \frac { 3 } { 4 } \right)$
C) $\left( 1 , \frac { 3 } { 4 } \right)$
D) $\left( - 1 , - \frac { 3 } { 4 } \right)$
E) $\left( 1 , \frac { 3 } { 2 } \right)$

Question 63

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, $\rho ( x , y ) = y ^ { 2 }$ and the y-axis with area density $\rho ( x , y ) = y ^ { 2 }$ is

A) $\left( - \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$
B) $\left( \frac { 6 } { 5 } , - \frac { 6 } { 5 } \right)$
C) $\left( \frac { 6 } { 5 } , - \frac { 9 } { 5 } \right)$
D) $\left( \frac { 6 } { 5 } , \frac { 6 } { 5 } \right)$
E) $\left( \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$

Question 64

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, x + y = 3 and the y-axis with area density $\rho ( x , y ) = x ^ { 2 } + y ^ { 2 }$ is

A) $\left( \frac { 6 } { 5 } , \frac { 6 } { 5 } \right)$
B) $\left( \frac { 27 } { 2 } , \frac { 79 } { 5 } \right)$
C) $\left( \frac { 23 } { 2 } , \frac { 81 } { 5 } \right)$
D) $\left( \frac { 25 } { 2 } , \frac { 81 } { 5 } \right)$
E) $\left( \frac { 21 } { 2 } , \frac { 81 } { 5 } \right)$

Question 65

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, y = x and y = 2 - x with area density $\rho ( x , y ) = 6 x + 3 y + 3$ is

A) $\left( \frac { 7 } { 8 } , \frac { 17 } { 16 } \right)$
B) $\left( \frac { 3 } { 8 } , \frac { 19 } { 16 } \right)$
C) $\left( \frac { 5 } { 8 } , \frac { 17 } { 16 } \right)$
D) $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$
E) $\left( \frac { 1 } { 8 } , \frac { 17 } { 16 } \right)$

Question 66

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, $y = x ^ { 2 }$ and y = 1 with area density $\rho ( x , y ) = x + y$ is

A) $\left( \frac { 27 } { 2 } , \frac { 81 } { 5 } \right)$
B) $\left( \frac { 6 } { 13 } , \frac { 190 } { 273 } \right)$
C) $\left( 2 , \frac { 3 } { 2 } \right)$
D) $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$
E) $\left( \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$

Question 67

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, $8 y = x ^ { 2 }$ and y = 2 with area density $\rho ( x , y ) = y + 1$ is

A) $\left( \frac { 27 } { 2 } , \frac { 81 } { 5 } \right)$
B) $\left( \frac { 6 } { 13 } , \frac { 190 } { 273 } \right)$
C) $\left( 2 , \frac { 3 } { 2 } \right)$
D) $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$
E) $\left( \frac { 35 } { 22 } , \frac { 102 } { 77 } \right)$

Question 68

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the $x ^ { 2 } + y ^ { 2 } = 1 , x \geq 0 , y \geq 0$ with area density $\rho ( x , y ) = x + y$ is

A) $\left( \frac { 3 ( 2 + \pi ) } { 16 } , \frac { 3 ( 2 + \pi ) } { 32 } \right)$
B) $\left( \frac { 3 ( 2 - \pi ) } { 32 } , \frac { 3 ( 2 - \pi ) } { 32 } \right)$
C) $\left( \frac { 3 ( 2 + \pi ) } { 32 } , \frac { 3 ( 2 + \pi ) } { 32 } \right)$
D) $\left( \frac { 3 ( 2 + \pi ) } { 32 } , \frac { 3 ( 2 - \pi ) } { 32 } \right)$
E) $\left( \frac { 3 ( 2 - \pi ) } { 32 } , \frac { 3 ( 2 + \pi ) } { 32 } \right)$

Question 69

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, $3 x + 2 y = 18$ and y = 0 with area density $\rho ( x , y ) = x y$ is

A) $\left( \frac { 27 } { 2 } , \frac { 81 } { 5 } \right)$
B) $\left( \frac { 12 } { 5 } , \frac { 18 } { 5 } \right)$
C) $\left( \frac { 3 ( 2 + \pi ) } { 32 } , \frac { 3 ( 2 + \pi ) } { 32 } \right)$
D) $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$
E) $\left( \frac { 35 } { 22 } , \frac { 102 } { 77 } \right)$

Question 70

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, $y = \sin x$ from x = 0 to x =?with area density $\rho ( x , y ) = y$ is

A) $\left( \frac { \pi } { 2 } , \frac { 9 } { 16 \pi } \right)$
B) $\left( \frac { \pi } { 2 } , \frac { 9 \pi } { 16 } \right)$
C) $\left( \frac { \pi } { 2 } , \frac { 16 \pi } { 7 } \right)$
D) $\left( \frac { \pi } { 2 } , \frac { 16 } { 9 \pi } \right)$
E) $\left( \frac { \pi } { 2 } , \frac { 16 \pi } { 9 } \right)$

Question 71

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $y = \sqrt { x }$ and y = x with area density $\rho ( x , y ) = x$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( \frac { 12 } { 5 } , \frac { 18 } { 5 } \right)$
C) $\left( \frac { 15 } { 28 } , \frac { 5 } { 9 } \right)$
D) $\left( \frac { 1 } { 2 } , \frac { 16 } { 9 } \right)$
E) $\left( \frac { 35 } { 22 } , \frac { 102 } { 77 } \right)$

Question 72

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 4 , x \geq 0 , y \geq 0$ and x + y = 2 with area density $\rho ( x , y ) = x y$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( \frac { 12 } { 5 } , \frac { 18 } { 5 } \right)$
C) $\left( \frac { 35 } { 22 } , \frac { 102 } { 77 } \right)$
D) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$
E) $\left( \frac { 6 } { 5 } , \frac { 6 } { 5 } \right)$

Question 73

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 1$ and y =1 with area density $\rho ( x , y ) = x y$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( \frac { 12 } { 5 } , \frac { 18 } { 5 } \right)$
C) $\left( \frac { 4 } { 5 } , \frac { 4 } { 5 } \right)$
D) $\left( \frac { 35 } { 22 } , \frac { 102 } { 77 } \right)$
E) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 74

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 ( 1 + \sin \theta )$ with area density $\rho ( r , \theta ) = r$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( 0 , \frac { 21 } { 10 } \right)$
C) $\left( \frac { 4 } { 5 } , \frac { 4 } { 5 } \right)$
D) $\left( \frac { 12 } { 5 } , \frac { 18 } { 5 } \right)$
E) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 75

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = \cos ( 2 \theta )$ for the petal on the right with area density $\rho ( r , \theta ) = r$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( 0 , \frac { 21 } { 10 } \right)$
C) $\left( \frac { 4 } { 5 } , \frac { 4 } { 5 } \right)$
D) $\left( \frac { 16 \sqrt { 2 } } { 35 } , 0 \right)$
E) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 76

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 - \cos ( \theta )$ with area density $\rho ( r , \theta ) = r$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( 0 , \frac { 21 } { 10 } \right)$
C) $\left( - \frac { 57 } { 44 } , 0 \right)$
D) $\left( \frac { 173 \sqrt { 2 } } { 280 } , 0 \right)$
E) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 77

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 + \cos ( \theta ) , 0 \leq \theta \leq \pi$ and the polar axis with area density $\rho ( r , \theta ) = \sin \theta$ is

A) $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$
B) $\left( 0 , \frac { 21 } { 10 } \right)$
C) $\left( - \frac { 57 } { 44 } , 0 \right)$
D) $\left( \frac { 42 } { 65 } , \frac { 19 \pi } { 52 } \right)$
E) $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 78

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $y = \frac { 3 } { 4 } x , x = 4$ and the x-axis with area density $\rho ( x , y ) = k$ about the x-axis is

A) $13 k$
B) $9 k$
C) $8 k$
D) $11 k$
E) $7 k$

Question 79

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 1 , y \geq 0$ and the x-axis with area density $\rho ( x , y ) = y$ about the x-axis is

A)13
B)9
C) $\frac { 65 } { 8 }$
D) $\frac { 4 } { 15 }$
E) $\frac { 1 } { 15 }$

Question 80

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x = 3 , y = 2 , x = 0$ and the x-axis with area density $\rho ( x , y ) = x y ^ { 2 }$ about the y-axis is

A)54
B)48
C)45
D)42
E)38