Question 1

Find the average value of f(x, y, z) = xy over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1.

Accepted Answer

The answer of Find the average value of f(x, y,...

Question 2

Suppose X and Y are random variables. Find k such that the function $f ( x , y ) = \left\{ \begin{array} { l l } 
k e ^ { - ( 0.6 x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \
0 & \text { otherwise }
\end{array} \right.$ is a joint density function.

Accepted Answer

A) 0.2 
B) 0.3 
C) 0.25 
D) 1 
E) 0.5
F)0.4
G)0.24
H)0 
A) 0.2 
B) 0.3 
C) 0.25 
D) 1 
E) 0.5
F)0.4
G)0.24
H)0

Question 3

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } e ^ { x ^ { 2 } } d x d y$ .

Accepted Answer

The answer of Evaluate the iterated integral \(\int _ {...

Question 4

Compute the Riemann sum for the double integral $\iint _ { R } x + 2 y d A$ where $R = [ 0,6 ] \times [ 0,2 ]$ for the given grid and choice of sample points.

Accepted Answer

The answer of Compute the Riemann sum for the double...

Question 5

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

Accepted Answer

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

Question 6

Use polar coordinates to find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of the line $x = 1$ .

Accepted Answer

The answer of Use polar coordinates to find the area...

Question 7

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

Accepted Answer

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

G) $\frac { 64 \pi } { 3 }$

H) $\frac { 128 \pi } { 3 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { 4 \pi } { 3 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 16 \pi } { 3 }$ 
F) $\frac { 32 \pi } { 3 }$

G) $\frac { 64 \pi } { 3 }$

H) $\frac { 128 \pi } { 3 }$

Question 8

Evaluate the double integral $\iint _ { R } ( x + \sin y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}$ .

Accepted Answer

A)  $2 + \pi$ 
B)  $4 + \pi$ 
C)  $( 2 + \pi ) / 2$ 
D)  $4 + 2 \pi$ 
E)  $- 2 - \pi$ 
F) $- 4 - \pi$

G) $- ( 2 + \pi ) / 2$

H) $- ( 4 + \pi ) / 2$ 
A)  $2 + \pi$ 
B)  $4 + \pi$ 
C)  $( 2 + \pi ) / 2$ 
D)  $4 + 2 \pi$ 
E)  $- 2 - \pi$ 
F) $- 4 - \pi$

G) $- ( 2 + \pi ) / 2$

H) $- ( 4 + \pi ) / 2$

Question 9

Use the Midpoint Rule to estimate $\iint _ { R } ( 7 - 2 x - y ) d A$ over $R = \{ ( x , y ) \mid - 1 \leq x \leq 2 , - 1 \leq y \leq 2 \}$ partitioned by the lines x = 0, x = 1, y = 0, and y = 1 into nine subrectangles.

Accepted Answer

The answer of Use the Midpoint Rule to estimate \(\iint...

Question 10

Find the area of the part of the paraboloid $x = y ^ { 2 } + z ^ { 2 }$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 9$ .

Accepted Answer

The answer of Find the area of the part of...

Question 11

Evaluate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { n / x } x ^ { 2 } \sin x y d y d x$ .

Accepted Answer

The answer of Evaluate the iterated integral \(\int _ {...

Question 12

Find the volume of the solid that is common to the cylinders $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ and $x ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ .

Accepted Answer

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

Question 13

Let E be the solid under the plane x + y + z = 5 and above the region in the xy-plane bounded by $x = 4 - y ^ { 2 }$ and x + y = 2. Express the volume of E as an iterated integral in rectangular coordinates.

Accepted Answer

The answer of Let E be the solid under the...

Question 14

Find the area of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Accepted Answer

The answer of Find the area of that part of...

Question 15

Find the 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) 0.35 
B) 0.40 
C) 0.45 
D) 0.50 
E) 0.55
F)0.60
G)0.65
H)0.70 
A) 0.35 
B) 0.40 
C) 0.45 
D) 0.50 
E) 0.55
F)0.60
G)0.65
H)0.70

Question 16

Describe the shape of the solid whose volume is given by the integral $\int _ { 0 } ^ { 3 } \int _ { - 2 } ^ { 2 } \sqrt { 9 - y ^ { 2 } } d x d y$ .

Accepted Answer

Quarter-cy

Question 17

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y } y d x d y d z$ .

Accepted Answer

A)  $\frac { 1 } { 120 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  $\frac { 1 } { 60 }$ 
D)  $\frac { 1 } { 48 }$ 
E)  $\frac { 1 } { 36 }$ 
F) $\frac { 1 } { 24 }$ 
 
G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$ 
A)  $\frac { 1 } { 120 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  $\frac { 1 } { 60 }$ 
D)  $\frac { 1 } { 48 }$ 
E)  $\frac { 1 } { 36 }$ 
F) $\frac { 1 } { 24 }$ 
 
G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 18

Find the area of the part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

Accepted Answer

The answer of Find the area of the part of...

Question 19

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

Accepted Answer

A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 4 }$ 
C) $\pi$ 
D) 2$\pi$ 
E)  $\frac { \pi } { 16 }$

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

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

H) $\frac { \pi } { 6 }$ 
A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 4 }$ 
C) $\pi$ 
D) 2$\pi$ 
E)  $\frac { \pi } { 16 }$

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

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

H) $\frac { \pi } { 6 }$

Question 20

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).

Accepted Answer

A) 0 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 1
F) $\frac { 3 } { 2 }$ 
G)2
H)-2 
A) 0 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 1
F) $\frac { 3 } { 2 }$ 
G)2
H)-2

Find the average value of f(x, y, z) = xy over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1.

Suppose X and Y are random variables. Find k such that the function $f ( x , y ) = \left\{ \begin{array} { l l } k e ^ { - ( 0.6 x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ is a joint density function.

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } e ^ { x ^ { 2 } } d x d y$ .

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

Use polar coordinates to find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of the line $x = 1$ .

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

Evaluate the double integral $\iint _ { R } ( x + \sin y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}$ .

Use the Midpoint Rule to estimate $\iint _ { R } ( 7 - 2 x - y ) d A$ over $R = \{ ( x , y ) \mid - 1 \leq x \leq 2 , - 1 \leq y \leq 2 \}$ partitioned by the lines x = 0, x = 1, y = 0, and y = 1 into nine subrectangles.

Find the area of the part of the paraboloid $x = y ^ { 2 } + z ^ { 2 }$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 9$ .

Evaluate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { n / x } x ^ { 2 } \sin x y d y d x$ .

Find the volume of the solid that is common to the cylinders $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ and $x ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ .

Let E be the solid under the plane x + y + z = 5 and above the region in the xy-plane bounded by $x = 4 - y ^ { 2 }$ and x + y = 2. Express the volume of E as an iterated integral in rectangular coordinates.

Find the area of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Find the 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.

Describe the shape of the solid whose volume is given by the integral $\int _ { 0 } ^ { 3 } \int _ { - 2 } ^ { 2 } \sqrt { 9 - y ^ { 2 } } d x d y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y } y d x d y d z$ .

Find the area of the part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

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

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).

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Exam 12: Multiple Integrals

Find the average value of f(x, y, z) = xy over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1.

Evaluate the iterated integral ∫01∫y1ex2dxdy\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } e ^ { x ^ { 2 } } d x d y∫01​∫y1​ex2dxdy .

Compute the Riemann sum for the double integral ∬Rx+2ydA\iint _ { R } x + 2 y d A∬R​x+2ydA where R=[0,6]×[0,2]R = [ 0,6 ] \times [ 0,2 ]R=[0,6]×[0,2] for the given grid and choice of sample points.

Use polar coordinates to find the area inside the circle x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 and to the right of the line x=1x = 1x=1 .

Evaluate the integral ∬Rx2+y2dA\iint _ { R } \sqrt { x ^ { 2 } + y ^ { 2 } } d A∬R​x2+y2​dA , where R is the disk with center the origin and radius 2.

Evaluate the double integral ∬R(x+sin⁡y)dA\iint _ { R } ( x + \sin y ) d A∬R​(x+siny)dA , where R={(x,y)∣0≤x≤2,0≤y≤π}R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}R={(x,y)∣0≤x≤2,0≤y≤π} .

Find the area of the part of the paraboloid x=y2+z2x = y ^ { 2 } + z ^ { 2 }x=y2+z2 that lies inside the cylinder y2+z2=9y ^ { 2 } + z ^ { 2 } = 9y2+z2=9 .

Evaluate the iterated integral ∫12∫0n/xx2sin⁡xydydx\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { n / x } x ^ { 2 } \sin x y d y d x∫12​∫0n/x​x2sinxydydx .

Find the volume of the solid that is common to the cylinders x2+y2=a2x ^ { 2 } + y ^ { 2 } = a ^ { 2 }x2+y2=a2 and x2+z2=a2x ^ { 2 } + z ^ { 2 } = a ^ { 2 }x2+z2=a2 .

Let E be the solid under the plane x + y + z = 5 and above the region in the xy-plane bounded by x=4−y2x = 4 - y ^ { 2 }x=4−y2 and x + y = 2. Express the volume of E as an iterated integral in rectangular coordinates.

Find the area of that part of the sphere x2+y2+z2=4zx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 zx2+y2+z2=4z that lies inside the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 .

Find the 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.

Describe the shape of the solid whose volume is given by the integral ∫03∫−229−y2dxdy\int _ { 0 } ^ { 3 } \int _ { - 2 } ^ { 2 } \sqrt { 9 - y ^ { 2 } } d x d y∫03​∫−22​9−y2​dxdy .

Evaluate the iterated integral ∫01∫0x∫0yydxdydz\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y } y d x d y d z∫01​∫0x​∫0y​ydxdydz .

Find the area of the part of the surface z=x+y2z = x + y ^ { 2 }z=x+y2 that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

Evaluate the iterated integral ∫02π∫01∫01−r2zrdzdrdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - r ^ { 2 } } } z r d z d r d \theta∫02π​∫01​∫01−r2​​zrdzdrdθ .

Evaluate the integral ∬Rx+yx−ydA\iint _ { R } \frac { x + y } { x - y } d A∬R​x−yx+y​dA , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).

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Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } e ^ { x ^ { 2 } } d x d y$ .

Use polar coordinates to find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of the line $x = 1$ .

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

Evaluate the double integral $\iint _ { R } ( x + \sin y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}$ .

Find the area of the part of the paraboloid $x = y ^ { 2 } + z ^ { 2 }$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 9$ .

Evaluate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { n / x } x ^ { 2 } \sin x y d y d x$ .

Find the volume of the solid that is common to the cylinders $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ and $x ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ .

Let E be the solid under the plane x + y + z = 5 and above the region in the xy-plane bounded by $x = 4 - y ^ { 2 }$ and x + y = 2. Express the volume of E as an iterated integral in rectangular coordinates.

Find the area of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Find the 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.

Describe the shape of the solid whose volume is given by the integral $\int _ { 0 } ^ { 3 } \int _ { - 2 } ^ { 2 } \sqrt { 9 - y ^ { 2 } } d x d y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y } y d x d y d z$ .

Find the area of the part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

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

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).