Question 1

Find the mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function p(x, y) = x.

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$ 
 
G) $\frac { 5 } { 6 }$

H)1 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$ 
 
G) $\frac { 5 } { 6 }$

H)1

Question 2

Suppose X and Y are random variables. Find k such that the function $f ( x , y ) = \left\{ \begin{array} { c l } 
0.1 e ^ { - ( \mid x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \
0 & \text { otherwise }
\end{array} \right.$ is a joint density function.

Accepted Answer

A) 0.2 
B) 2 
C) 0.25 
D) 1 
E) 2.5
F)0.4
G)0.1
H)0 
A) 0.2 
B) 2 
C) 0.25 
D) 1 
E) 2.5
F)0.4
G)0.1
H)0

Question 3

Compute the average value of $f ( x , y ) = 3 + x y ^ { 2 }$ over $R = [ - 2,2 ] \times [ 0,1 ]$ .

Accepted Answer

The answer of Compute the average value of \(f (...

Question 4

Find the x-coordinate of the center of mass of the lamina that occupies the region $D = \left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq 1 \right\}$ and has density function p(x, y) = x + y.

Accepted Answer

A)  $\frac { 6 } { 13 }$ 
B)  $\frac { 7 } { 13 }$ 
C)  $\frac { 8 } { 13 }$ 
D)  $\frac { 9 } { 13 }$ 
E)  $\frac { 6 } { 11 }$  
F) $\frac { 7 } { 11 }$

G) $\frac { 8 } { 11 }$

H) $\frac { 9 } { 11 }$ 
A)  $\frac { 6 } { 13 }$ 
B)  $\frac { 7 } { 13 }$ 
C)  $\frac { 8 } { 13 }$ 
D)  $\frac { 9 } { 13 }$ 
E)  $\frac { 6 } { 11 }$  
F) $\frac { 7 } { 11 }$

G) $\frac { 8 } { 11 }$

H) $\frac { 9 } { 11 }$

Question 5

Evaluate the iterated integral $\int _ { 2 } ^ { 5 } \int _ { 0 } ^ { 2 } x \sqrt { 4 - x ^ { 2 } } d x d y$ .

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 6 
E) 9
F)8
G)18
H)24 
A) 2 
B) 3 
C) 4 
D) 6 
E) 9
F)8
G)18
H)24

Question 6

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Accepted Answer

A) 3$\pi$ 
B) 2$\pi$ 
C) 4$\pi$ 
D) 8$\pi$ 
E) 12$\pi$
F)6$\pi$
G)16$\pi$
H)8 
A) 3$\pi$ 
B) 2$\pi$ 
C) 4$\pi$ 
D) 8$\pi$ 
E) 12$\pi$
F)6$\pi$
G)16$\pi$
H)8

Question 7

Evaluate the iterated integral $\int _ { 0 } ^ { 2 x } \int _ { 0 } ^ { 1 } \int _ { 1 - y ^ { 2 } } ^ { 4 } r ^ { 2 } d z d r d \theta$ .

Accepted Answer

A)  $\frac { \pi } { 5 }$ 
B)  $\frac { 7 \pi } { 5 }$ 
C) 4$\pi$ 
D) 12$\pi$ 
E)  $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { 12 \pi } { 3 }$

H) $\frac { 12 \pi } { 5 }$ 
A)  $\frac { \pi } { 5 }$ 
B)  $\frac { 7 \pi } { 5 }$ 
C) 4$\pi$ 
D) 12$\pi$ 
E)  $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { 12 \pi } { 3 }$

H) $\frac { 12 \pi } { 5 }$

Question 8

Evaluate the double integral $\iint _ { R } y \cos ( x y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq \pi \}$ .

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)4 
A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)4

Question 9

Calculate the double integral $\iint _ { R } x y ^ { 2 } + \frac { y } { x } d A$ , where $R = \{ ( x , y ) \mid 2 \leq x \leq 3 , - 1 \leq y \leq 0 \}$ .

Accepted Answer

The answer of Calculate the double integral \(\iint _ {...

Question 10

Evaluate $\iint _ { R } ( x + y ) e ^ { x ^ { 2 } - y ^ { 2 } } d A$ ,where R is the rectangular region bounded by the lines x + y = 0, x + y = 1, x - y = 0, and x - y = 1.

Accepted Answer

The answer of Evaluate \(\iint _ { R } (...

Question 11

The joint density function for a pair of random variables X and Y is $f ( x , y ) = \left\{ \begin{array} { l l } 
C \frac { x } { y } & \text { if } 0 \leq x \leq 1 \text { and } 1 \leq y \leq 2 \
0 & \text { otherwise }
\end{array} \right.$ . Find the value of C.

Accepted Answer

The answer of The joint density function for a pair...

Question 12

Electric charge is distributed over the unit disk $x ^ { 2 } + y ^ { 2 } \leq 1$ so that the charge density at (x, y) is $\sigma ( x , y ) = 1 + x ^ { 2 } + y ^ { 2 }$ (measured in coulombs per square meter). Find the total charge on the disk.

Accepted Answer

The answer of Electric charge is distributed over the unit...

Question 13

Find the volume of the solid bounded by the paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 2 - x ^ { 2 } - y ^ { 2 }$ .

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 14

Suppose X and Y are random variables whose density function is given by $f ( x , y ) = \left\{ \begin{array} { l l } 
0.2 e ^ { - ( 0.5 x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \
0 & \text { otherwise }
\end{array} \right.$ . Find the probability $P ( X \leq 1 , Y \leq 2 )$ .

Accepted Answer

The answer of Suppose X and Y are random variables...

Question 15

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ above the plane z = 2.

Accepted Answer

A) 3$\pi$ 
B) 2$\pi$ 
C) 4$\pi$ 
D) 8$\pi$ 
E) 12$\pi$
F)6$\pi$
G)16$\pi$
H)8 
A) 3$\pi$ 
B) 2$\pi$ 
C) 4$\pi$ 
D) 8$\pi$ 
E) 12$\pi$
F)6$\pi$
G)16$\pi$
H)8

Question 16

Evaluate the integral $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where R is the disk with center the origin and radius 1.

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
F) $\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H) $\pi$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
F) $\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H) $\pi$

Question 17

Set up, but do not evaluate, the integral to find the surface area of the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Accepted Answer

The answer of Set up, but do not evaluate, the...

Question 18

Find the volume under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ above the region bounded by the x-axis, the y-axis, and the line x + y = 1.

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Question 19

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$ .

Accepted Answer

A)  $\frac { 9 } { 8 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \ln \frac { 1 } { 2 }$ 
D)  $\ln \frac { 9 } { 8 }$ 
E)  $\frac { 8 } { 9 }$ 
F) $\frac { 3 } { 2 }$

G) $\ln \frac { 3 } { 2 }$

H) $\ln \frac { 8 } { 9 }$ 
A)  $\frac { 9 } { 8 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \ln \frac { 1 } { 2 }$ 
D)  $\ln \frac { 9 } { 8 }$ 
E)  $\frac { 8 } { 9 }$ 
F) $\frac { 3 } { 2 }$

G) $\ln \frac { 3 } { 2 }$

H) $\ln \frac { 8 } { 9 }$

Question 20

Find the area of the part of the plane x + z = 4 that lies above the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1).

Accepted Answer

A) 1 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) 2 
E)  $\sqrt { 5 }$ 
F) $\sqrt { 6 }$  
G) $\frac { \sqrt { 5 } } { 2 }$ 
H) $\frac { \sqrt { 6 } } { 2 }$ 
A) 1 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) 2 
E)  $\sqrt { 5 }$ 
F) $\sqrt { 6 }$  
G) $\frac { \sqrt { 5 } } { 2 }$ 
H) $\frac { \sqrt { 6 } } { 2 }$

Find the mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function p(x, y) = x.

Suppose X and Y are random variables. Find k such that the function $f ( x , y ) = \left\{ \begin{array} { c l } 0.1 e ^ { - ( \mid x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ is a joint density function.

Compute the average value of $f ( x , y ) = 3 + x y ^ { 2 }$ over $R = [ - 2,2 ] \times [ 0,1 ]$ .

Find the x-coordinate of the center of mass of the lamina that occupies the region $D = \left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq 1 \right\}$ and has density function p(x, y) = x + y.

Evaluate the iterated integral $\int _ { 2 } ^ { 5 } \int _ { 0 } ^ { 2 } x \sqrt { 4 - x ^ { 2 } } d x d y$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Evaluate the iterated integral $\int _ { 0 } ^ { 2 x } \int _ { 0 } ^ { 1 } \int _ { 1 - y ^ { 2 } } ^ { 4 } r ^ { 2 } d z d r d \theta$ .

Evaluate the double integral $\iint _ { R } y \cos ( x y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq \pi \}$ .

Calculate the double integral $\iint _ { R } x y ^ { 2 } + \frac { y } { x } d A$ , where $R = \{ ( x , y ) \mid 2 \leq x \leq 3 , - 1 \leq y \leq 0 \}$ .

Evaluate $\iint _ { R } ( x + y ) e ^ { x ^ { 2 } - y ^ { 2 } } d A$ ,where R is the rectangular region bounded by the lines x + y = 0, x + y = 1, x - y = 0, and x - y = 1.

The joint density function for a pair of random variables X and Y is $f ( x , y ) = \left\{ \begin{array} { l l } C \frac { x } { y } & \text { if } 0 \leq x \leq 1 \text { and } 1 \leq y \leq 2 \\0 & \text { otherwise }\end{array} \right.$ . Find the value of C.

Electric charge is distributed over the unit disk $x ^ { 2 } + y ^ { 2 } \leq 1$ so that the charge density at (x, y) is $\sigma ( x , y ) = 1 + x ^ { 2 } + y ^ { 2 }$ (measured in coulombs per square meter). Find the total charge on the disk.

Find the volume of the solid bounded by the paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 2 - x ^ { 2 } - y ^ { 2 }$ .

Suppose X and Y are random variables whose density function is given by $f ( x , y ) = \left\{ \begin{array} { l l } 0.2 e ^ { - ( 0.5 x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ . Find the probability $P ( X \leq 1 , Y \leq 2 )$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ above the plane z = 2.

Evaluate the integral $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where R is the disk with center the origin and radius 1.

Set up, but do not evaluate, the integral to find the surface area of the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Find the volume under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ above the region bounded by the x-axis, the y-axis, and the line x + y = 1.

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$ .

Find the area of the part of the plane x + z = 4 that lies above the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1).

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Exam 12: Multiple Integrals

Find the mass of the lamina that occupies the region D={(x,y)∣0≤x≤1,0≤y≤1}D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}D={(x,y)∣0≤x≤1,0≤y≤1} and has density function p(x, y) = x.

Compute the average value of f(x,y)=3+xy2f ( x , y ) = 3 + x y ^ { 2 }f(x,y)=3+xy2 over R=[−2,2]×[0,1]R = [ - 2,2 ] \times [ 0,1 ]R=[−2,2]×[0,1] .

Find the x-coordinate of the center of mass of the lamina that occupies the region D={(x,y)∣0≤x≤1,x2≤y≤1}D = \left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq 1 \right\}D={(x,y)∣0≤x≤1,x2≤y≤1} and has density function p(x, y) = x + y.

Evaluate the iterated integral ∫25∫02x4−x2dxdy\int _ { 2 } ^ { 5 } \int _ { 0 } ^ { 2 } x \sqrt { 4 - x ^ { 2 } } d x d y∫25​∫02​x4−x2​dxdy .

Find the area of the part of the sphere x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16 between the planes z = 1 and z = 2.

Evaluate the iterated integral ∫02x∫01∫1−y24r2dzdrdθ\int _ { 0 } ^ { 2 x } \int _ { 0 } ^ { 1 } \int _ { 1 - y ^ { 2 } } ^ { 4 } r ^ { 2 } d z d r d \theta∫02x​∫01​∫1−y24​r2dzdrdθ .

Evaluate the double integral ∬Rycos⁡(xy)dA\iint _ { R } y \cos ( x y ) d A∬R​ycos(xy)dA , where R={(x,y)∣0≤x≤1,0≤y≤π}R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq \pi \}R={(x,y)∣0≤x≤1,0≤y≤π} .

Calculate the double integral ∬Rxy2+yxdA\iint _ { R } x y ^ { 2 } + \frac { y } { x } d A∬R​xy2+xy​dA , where R={(x,y)∣2≤x≤3,−1≤y≤0}R = \{ ( x , y ) \mid 2 \leq x \leq 3 , - 1 \leq y \leq 0 \}R={(x,y)∣2≤x≤3,−1≤y≤0} .

Evaluate ∬R(x+y)ex2−y2dA\iint _ { R } ( x + y ) e ^ { x ^ { 2 } - y ^ { 2 } } d A∬R​(x+y)ex2−y2dA ,where R is the rectangular region bounded by the lines x + y = 0, x + y = 1, x - y = 0, and x - y = 1.

Electric charge is distributed over the unit disk x2+y2≤1x ^ { 2 } + y ^ { 2 } \leq 1x2+y2≤1 so that the charge density at (x, y) is σ(x,y)=1+x2+y2\sigma ( x , y ) = 1 + x ^ { 2 } + y ^ { 2 }σ(x,y)=1+x2+y2 (measured in coulombs per square meter). Find the total charge on the disk.

Find the volume of the solid bounded by the paraboloids z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z=2−x2−y2z = 2 - x ^ { 2 } - y ^ { 2 }z=2−x2−y2 .

Find the area of the part of the sphere x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16 above the plane z = 2.

Evaluate the integral ∬R(x2+y2)dA\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A∬R​(x2+y2)dA , where R is the disk with center the origin and radius 1.

Set up, but do not evaluate, the integral to find the surface area of the portion of the sphere x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16 between the planes z = 1 and z = 2.

Find the volume under the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 above the region bounded by the x-axis, the y-axis, and the line x + y = 1.

Evaluate the double integral ∬Ry(x+Y)2dA\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A∬R​(x+Y)2y​dA , where R={(x,y)∣1≤x≤2,1≤y≤2}R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}R={(x,y)∣1≤x≤2,1≤y≤2} .

Find the area of the part of the plane x + z = 4 that lies above the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1).

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Find the mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function p(x, y) = x.

Compute the average value of $f ( x , y ) = 3 + x y ^ { 2 }$ over $R = [ - 2,2 ] \times [ 0,1 ]$ .

Find the x-coordinate of the center of mass of the lamina that occupies the region $D = \left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq 1 \right\}$ and has density function p(x, y) = x + y.

Evaluate the iterated integral $\int _ { 2 } ^ { 5 } \int _ { 0 } ^ { 2 } x \sqrt { 4 - x ^ { 2 } } d x d y$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Evaluate the iterated integral $\int _ { 0 } ^ { 2 x } \int _ { 0 } ^ { 1 } \int _ { 1 - y ^ { 2 } } ^ { 4 } r ^ { 2 } d z d r d \theta$ .

Evaluate the double integral $\iint _ { R } y \cos ( x y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq \pi \}$ .

Calculate the double integral $\iint _ { R } x y ^ { 2 } + \frac { y } { x } d A$ , where $R = \{ ( x , y ) \mid 2 \leq x \leq 3 , - 1 \leq y \leq 0 \}$ .

Evaluate $\iint _ { R } ( x + y ) e ^ { x ^ { 2 } - y ^ { 2 } } d A$ ,where R is the rectangular region bounded by the lines x + y = 0, x + y = 1, x - y = 0, and x - y = 1.

Electric charge is distributed over the unit disk $x ^ { 2 } + y ^ { 2 } \leq 1$ so that the charge density at (x, y) is $\sigma ( x , y ) = 1 + x ^ { 2 } + y ^ { 2 }$ (measured in coulombs per square meter). Find the total charge on the disk.

Find the volume of the solid bounded by the paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 2 - x ^ { 2 } - y ^ { 2 }$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ above the plane z = 2.

Evaluate the integral $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where R is the disk with center the origin and radius 1.

Set up, but do not evaluate, the integral to find the surface area of the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ between the planes z = 1 and z = 2.

Find the volume under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ above the region bounded by the x-axis, the y-axis, and the line x + y = 1.

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$ .