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Deck 8: Calculus of Several Variables
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Question
Find the volume of the solid bounded above by the surface 
and below by the plane region R. 
; R is the region bounded by 
.
and below by the plane region R. ; R is the region bounded by .
Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function 
Where
stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per week
B)
per week
C)
per week
D)
per week
E)
per week
Where
stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per weekB)
per weekC)
per weekD)
per weekE)
per week Question
Find the average value of the given function 
over the plane region R. 
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A) 
B) 
C) 
D) 
E) 
over the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
E)
Question
Find the volume of the solid bounded above by the surfac 
and below by the plane region R. 
; R is the triangle with vertices (0, 0), (2, 0) and (0, 2).
and below by the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2). Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 36
B) 24
C) 144
D) 12
E) 48
A) 36
B) 24
C) 144
D) 12
E) 48
Question
Find the volume of the solid bounded above by the surface 
and below by the plane region R. 
and R is the region bounded by the graphs of 
.
A)
B) 
C) 
D) 
E) 
and below by the plane region R. and R is the region bounded by the graphs of .
A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the solid shown in the figure.
__________ cu units.
__________ cu units.
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the volume of the solid bounded above by the surface 
and below by the plane region R. 
and R is the region bounded by the graphs of 
and 
.
and below by the plane region R. and R is the region bounded by the graphs of and . Question
The population density of a certain city is given by the function 
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B) 
C) 
D) 
E) 
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the average value of the function 
over the plane region R. 
and R is the triangle with vertices 
, 
and 
.
A) 5
B) 
C) 25
D) 
E)
over the plane region R. and R is the triangle with vertices , and . A) 5
B)
C) 25
D)
E)
Question
Find the volume of the solid bounded above by the surface 
And below by the plane region R.
R is the triangle with vertices
and 
.
A)
B) 
C) 
D) 
E) 
And below by the plane region R.
R is the triangle with vertices
and .
A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 12
B) 96
C) 64
D) 256
E) 384
A) 12
B) 96
C) 64
D) 256
E) 384
Question
Find the volume of the solid bounded above by the surface 
and below by the plane region R. 
; R is the region bounded by 
.
A)
B) 
C) 
D) 
E) 
and below by the plane region R. ; R is the region bounded by .
A)
B)
C)
D)
E)
Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
Question
Use a double integral to find the volume of the solid shown in the figure.
__________ cu units.
__________ cu units.
Question
Find the average value of the given function 
over the plane region R. 
; R is the region bounded by the graph of 
and 
from 
to 
.
A)
B) 
C) 
D) 
E) 
over the plane region R. ; R is the region bounded by the graph of and from to .
A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the solid shown in the figure.
Question
Use a double integral to find the volume of the solid shown in the figure. 
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
F(x, y) = 6x + 6y; R is bounded by x = 0,
, y = 0 and y = 4.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 6x + 6y; R is bounded by x = 0,
, y = 0 and y = 4.
A)
B)
C)
D)
Question
Find the volume of the solid bounded above by the surface
Z = f(x, y)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
Z = f(x, y)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
Question
Evaluate the double integral 
For the function
and the region R.
and R is the rectangle defined by 
and 
.
A) 8
B) 6
C) 2
D) 3
E) 5
For the function
and the region R.
and R is the rectangle defined by and .
A) 8
B) 6
C) 2
D) 3
E) 5
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
F(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
; R is the rectangle defined by 
and 
.
A) 0
B) 4
C) -4
D) 1
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) 0
B) 4
C) -4
D) 1
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B)
C)
D)
A)
B)
C)
D)
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
; R is bounded by 
and y = x.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
; R is bounded by and y = x.
A)
B)
C)
D)
Question
Find the average value of the given function 
over the plane region R.
; R is the triangle with vertices (0, 0), (1, 0) and (1, 1).
over the plane region R.
; R is the triangle with vertices (0, 0), (1, 0) and (1, 1). Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
F(x, y) = 2y + x; R is the rectangle defined by
and 
.
A)
B)
C)
D) 26
For the given function f(x, y) and the region R.
F(x, y) = 2y + x; R is the rectangle defined by
and .
A)
B)
C)
D) 26
Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 6
B) 4
C) 9
D) 14
A) 6
B) 4
C) 9
D) 14
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 2.5
B) 3.5e
C) -3.5
D) -2.5
For the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 2.5
B) 3.5e
C) -3.5
D) -2.5
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
Question
Find the average value of the given function 
over the plane region R.
; R is the region bounded by the graph of 
and 
from 
to 
.
over the plane region R.
; R is the region bounded by the graph of and from to . Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
; R is the rectangle defined by 
and 
A)
B)
C)
D)
For the given function f(x, y) and the region R.
; R is the rectangle defined by and A)
B)
C)
D)
Question
Find the volume of the solid bounded above by the surface
Z = f(x, y)
And below by the plane region R.
; R is the region bounded by 
, 
, and 
.
A)
B)
C)
Z = f(x, y)
And below by the plane region R.
; R is the region bounded by , , and .
A)
B)
C)
Question
Evaluate the double integral 
For the function
and the region R.
and R is bounded by 
, 
, 
and 
.
A)
B)
C)
D)
E)
For the function
and the region R.
and R is bounded by , , and .
A)
B)
C)
D)
E)
Question
Evaluate the double integral 
For the given function f(x, y) and the region R.
; R is the rectangle defined by 
and 
.
A) -10
B) 10e
C) 9
D) -9e
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) -10
B) 10e
C) 9
D) -9e
Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 5,996
B) 6,008
C) 6,000
D) 5,994
A) 5,996
B) 6,008
C) 6,000
D) 5,994
Question
Find the average value of the function 
over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1). Question
Find the volume of the solid bounded above by the surface 
And below by the plane region R.
and R is the region bounded by the graphs of 
and 
.
A)
B)
C)
D)
E)
And below by the plane region R.
and R is the region bounded by the graphs of and .
A)
B)
C)
D)
E)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. Enter your answer as a fraction. 
; R is the rectangle defined by 
and 
.
for the given function f(x, y) and the region R. Enter your answer as a fraction. ; R is the rectangle defined by and . Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function 
Where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per week
B)
per week
C)
per week
D)
per week
E)
per week
Where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per weekB)
per weekC)
per weekD)
per weekE)
per week Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_6725_b0a1_abb0ce5e996b_TB8255_11.jpg)
where ![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_6726_b0a1_05e29c894151_TB8255_11.jpg)
.
A) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_6727_b0a1_2b5f95c919b1_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e38_b0a1_93acc4ae3fb9_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e39_b0a1_01db2b4e5c74_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3a_b0a1_e99c7fc18596_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3b_b0a1_c100a560b7f6_TB8255_11.jpg)
, so the statement is false.
B) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3c_b0a1_3b81855ebcad_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3d_b0a1_9f958824f5c9_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3e_b0a1_e5aea5cc6ad2_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e3f_b0a1_b9cefa9d035c_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_8e40_b0a1_bf28c28213bb_TB8255_11.jpg)
, so the statement is true.
C) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_b551_b0a1_6deebf5c8047_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_b552_b0a1_0d06b43c908a_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_b553_b0a1_f57b73363d8b_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_b554_b0a1_153e0fa2a49d_TB8255_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where . </strong> A) From the definition follows , so the statement is false. B) From the definition follows , so the statement is true. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cadb_b555_b0a1_b94a8229bff6_TB8255_11.jpg)
, so the statement is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where .
A) From the definition follows
, so the statement is false.B) From the definition follows
, so the statement is true.C) From the definition follows
, so the statement is false. Question
Use a double integral to find the volume of the solid shown in the figure.
Question
Find the average value of the function 
Over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D)
E)
Over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D)
E)
Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
Question
Evaluate the double integral
for the given function f(x, y) and the region R. Enter your answer as a fraction.
f(x, y) = 0x + 0y; R is bounded by x = 0,
, y = 0 and y = 4.
for the given function f(x, y) and the region R. Enter your answer as a fraction.
f(x, y) = 0x + 0y; R is bounded by x = 0,
, y = 0 and y = 4. Question
Evaluate the double integral
for the function
and the region R.
and R is the rectangle defined by 
and 
.
for the function
and the region R.
and R is the rectangle defined by and . Question
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is the rectangle defined by 
and 
.
for the given function f(x, y) and the region R.
; R is the rectangle defined by and . Question
Evaluate the double integral 
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 2y + 3x; R is the rectangle defined by 
and 
.
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 2y + 3x; R is the rectangle defined by and . Question
Find the average value of the given function 
Over the plane region R.
; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7.
A)
B)
C)
D)
Over the plane region R.
; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7.
A)
B)
C)
D)
Question
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x. Question
Evaluate the double integral 
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x. Question
Use a double integral to find the volume of the solid shown in the figure.
Question
The population density of a certain city is given by the function 
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1. Question
Find the average value of the given function
F(x, y)
Over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
F(x, y)
Over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
Question
Evaluate the double integral
for the function
and the region R. Enter your answer as an expression.
and R is bounded by 
, 
, 
and 
.
for the function
and the region R. Enter your answer as an expression.
and R is bounded by , , and . Question
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is the rectangle defined by 
and 
.
for the given function f(x, y) and the region R.
; R is the rectangle defined by and . Question
Find the total differential of the function. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the total differential of the function. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the approximate change in z when the point 
changes from 
to 
. 
; from 
to 
A) 
B) 
C) 
D) 
E)
changes from to . ; from to A)
B)
C)
D)
E)
Question
Find the average value of the given function
f(x, y)
over the plane region R. Enter your answer as a fraction.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).
f(x, y)
over the plane region R. Enter your answer as a fraction.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5). Question
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the region bounded by 
, 
and 
.
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the region bounded by , and . Question
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the triangle with vertices (0, 0), (4, 0) and (0, 4). Question
Use a double integral to find the volume of the solid shown in the figure. Enter your answer as an expression.
Question
Find the total differential of the function. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
![Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where .<div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cade_26a1_b0a1_652ca1a399ab_TB8255_11.jpg)
where ![Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where .<div style=padding-top: 35px>](https://d2lvgg3v3hfg70.cloudfront.net/TB8255/11eb651f_cade_26a2_b0a1_bd6e5e1559fb_TB8255_11.jpg)
.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where . Question
Find the approximate change in z when the point 
changes from 
to 
. 
; from 
to 
A) 
B) 
C) 
D) 
E)
changes from to . ; from to A)
B)
C)
D)
E)
Question
Find the approximate change in z when the point 
changes from 
to 
. 
; from 
to 
A) 
B)
C) 
D) 
E) 
changes from to . ; from to A)
B)
C)
D)
E)
Question
Evaluate the double integral
for the given function f(x, y) and the region R. Enter your answer as an expression.
; R is bounded by 
and y = x.
for the given function f(x, y) and the region R. Enter your answer as an expression.
; R is bounded by and y = x. Question
Find the average value of the given function
f(x, y)
over the plane region R. Enter your answer as an expression.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 7.
f(x, y)
over the plane region R. Enter your answer as an expression.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 7. Question
Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula.
Question
Find the total differential of the function. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the total differential of the function. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the average value of the function
over the plane region R. Enter your answer as an expression.
and R is the triangle with vertices 
, 
and 
.
over the plane region R. Enter your answer as an expression.
and R is the triangle with vertices , and . Question
Find the volume of the solid bounded above by the surface
and below by the plane region R. Enter your answer as a formula.
and R is the region bounded by the graphs of 
and 
.
and below by the plane region R. Enter your answer as a formula.
and R is the region bounded by the graphs of and . Question
The population density of a certain city is given by the function
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
. Enter your answer as an expression.
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
. Enter your answer as an expression.
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Deck 8: Calculus of Several Variables
1
Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .
and below by the plane region R. ; R is the region bounded by .
2
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
Where stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A) per week
B) per week
C) per week
D) per week
E) per week
Where
stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per weekB)
per weekC)
per weekD)
per weekE)
per week per week 3
Find the average value of the given function over the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
E)
over the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
E)
4
Find the volume of the solid bounded above by the surfac and below by the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2).
and below by the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2). Unlock Deck
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5
Use a double integral to find the volume of the solid shown in the figure.
A) 36
B) 24
C) 144
D) 12
E) 48
A) 36
B) 24
C) 144
D) 12
E) 48
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6
Find the volume of the solid bounded above by the surface and below by the plane region R. and R is the region bounded by the graphs of .
A)
B)
C)
D)
E)
and below by the plane region R. and R is the region bounded by the graphs of .
A)
B)
C)
D)
E)
Unlock Deck
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7
Use a double integral to find the volume of the solid shown in the figure.
__________ cu units.
__________ cu units.
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8
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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9
Find the volume of the solid bounded above by the surface and below by the plane region R. and R is the region bounded by the graphs of and .
and below by the plane region R. and R is the region bounded by the graphs of and . Unlock Deck
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10
The population density of a certain city is given by the function
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
E)
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
11
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Unlock Deck
k this deck
12
Find the average value of the function over the plane region R. and R is the triangle with vertices , and .
A) 5
B)
C) 25
D)
E)
over the plane region R. and R is the triangle with vertices , and . A) 5
B)
C) 25
D)
E)
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13
Find the volume of the solid bounded above by the surface
And below by the plane region R.
R is the triangle with vertices and .
A)
B)
C)
D)
E)
And below by the plane region R.
R is the triangle with vertices
and .
A)
B)
C)
D)
E)
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14
Use a double integral to find the volume of the solid shown in the figure.
A) 12
B) 96
C) 64
D) 256
E) 384
A) 12
B) 96
C) 64
D) 256
E) 384
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k this deck
15
Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .
A)
B)
C)
D)
E)
and below by the plane region R. ; R is the region bounded by .
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
16
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
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17
Use a double integral to find the volume of the solid shown in the figure.
__________ cu units.
__________ cu units.
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18
Find the average value of the given function over the plane region R. ; R is the region bounded by the graph of and from to .
A)
B)
C)
D)
E)
over the plane region R. ; R is the region bounded by the graph of and from to .
A)
B)
C)
D)
E)
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19
Use a double integral to find the volume of the solid shown in the figure.
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20
Use a double integral to find the volume of the solid shown in the figure.
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21
Evaluate the double integral
For the given function f(x, y) and the region R.
F(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 6x + 6y; R is bounded by x = 0,
, y = 0 and y = 4.
A)
B)
C)
D)
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22
Find the volume of the solid bounded above by the surface
Z = f(x, y)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
Z = f(x, y)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
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23
Evaluate the double integral
For the function and the region R.
and R is the rectangle defined by and .
A) 8
B) 6
C) 2
D) 3
E) 5
For the function
and the region R.
and R is the rectangle defined by and .
A) 8
B) 6
C) 2
D) 3
E) 5
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24
Evaluate the double integral
For the given function f(x, y) and the region R.
F(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
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25
Evaluate the double integral
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) 0
B) 4
C) -4
D) 1
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) 0
B) 4
C) -4
D) 1
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26
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
A)
B)
C)
D)
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Unlock Deck
k this deck
27
Evaluate the double integral
For the given function f(x, y) and the region R.
; R is bounded by and y = x.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
; R is bounded by and y = x.
A)
B)
C)
D)
Unlock Deck
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Unlock Deck
k this deck
28
Find the average value of the given function over the plane region R.
; R is the triangle with vertices (0, 0), (1, 0) and (1, 1).
over the plane region R.
; R is the triangle with vertices (0, 0), (1, 0) and (1, 1). Unlock Deck
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29
Evaluate the double integral
For the given function f(x, y) and the region R.
F(x, y) = 2y + x; R is the rectangle defined by and .
A)
B)
C)
D) 26
For the given function f(x, y) and the region R.
F(x, y) = 2y + x; R is the rectangle defined by
and .
A)
B)
C)
D) 26
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30
Use a double integral to find the volume of the solid shown in the figure.
A) 6
B) 4
C) 9
D) 14
A) 6
B) 4
C) 9
D) 14
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31
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Unlock Deck
k this deck
32
Evaluate the double integral
For the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 2.5
B) 3.5e
C) -3.5
D) -2.5
For the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 2.5
B) 3.5e
C) -3.5
D) -2.5
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33
Evaluate the double integral
For the given function f(x, y) and the region R.
F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
For the given function f(x, y) and the region R.
F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
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34
Find the average value of the given function over the plane region R.
; R is the region bounded by the graph of and from to .
over the plane region R.
; R is the region bounded by the graph of and from to . Unlock Deck
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35
Evaluate the double integral
For the given function f(x, y) and the region R.
; R is the rectangle defined by and
A)
B)
C)
D)
For the given function f(x, y) and the region R.
; R is the rectangle defined by and A)
B)
C)
D)
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k this deck
36
Find the volume of the solid bounded above by the surface
Z = f(x, y)
And below by the plane region R.
; R is the region bounded by , , and .
A)
B)
C)
Z = f(x, y)
And below by the plane region R.
; R is the region bounded by , , and .
A)
B)
C)
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37
Evaluate the double integral
For the function and the region R.
and R is bounded by , , and .
A)
B)
C)
D)
E)
For the function
and the region R.
and R is bounded by , , and .
A)
B)
C)
D)
E)
Unlock Deck
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Unlock Deck
k this deck
38
Evaluate the double integral
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) -10
B) 10e
C) 9
D) -9e
For the given function f(x, y) and the region R.
; R is the rectangle defined by and .
A) -10
B) 10e
C) 9
D) -9e
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39
Use a double integral to find the volume of the solid shown in the figure.
A) 5,996
B) 6,008
C) 6,000
D) 5,994
A) 5,996
B) 6,008
C) 6,000
D) 5,994
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40
Find the average value of the function over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1). Unlock Deck
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k this deck
41
Find the volume of the solid bounded above by the surface
And below by the plane region R.
and R is the region bounded by the graphs of and .
A)
B)
C)
D)
E)
And below by the plane region R.
and R is the region bounded by the graphs of and .
A)
B)
C)
D)
E)
Unlock Deck
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42
Evaluate the double integral for the given function f(x, y) and the region R. Enter your answer as a fraction. ; R is the rectangle defined by and .
for the given function f(x, y) and the region R. Enter your answer as a fraction. ; R is the rectangle defined by and . Unlock Deck
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Unlock Deck
k this deck
43
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
Where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A) per week
B) per week
C) per week
D) per week
E) per week
Where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
A)
per weekB)
per weekC)
per weekD)
per weekE)
per week Unlock Deck
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k this deck
44
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where .
A) From the definition follows , so the statement is false.
B) From the definition follows , so the statement is true.
C) From the definition follows , so the statement is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where .
A) From the definition follows
, so the statement is false.B) From the definition follows
, so the statement is true.C) From the definition follows
, so the statement is false. Unlock Deck
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45
Use a double integral to find the volume of the solid shown in the figure.
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46
Find the average value of the function
Over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D)
E)
Over the plane region R.
and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D)
E)
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k this deck
47
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
where x stands for the number of finished units and y stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
P = $__________ per week.
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48
Evaluate the double integral
for the given function f(x, y) and the region R. Enter your answer as a fraction.
f(x, y) = 0x + 0y; R is bounded by x = 0, , y = 0 and y = 4.
for the given function f(x, y) and the region R. Enter your answer as a fraction.
f(x, y) = 0x + 0y; R is bounded by x = 0,
, y = 0 and y = 4. Unlock Deck
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k this deck
49
Evaluate the double integral
for the function and the region R.
and R is the rectangle defined by and .
for the function
and the region R.
and R is the rectangle defined by and . Unlock Deck
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50
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is the rectangle defined by and .
for the given function f(x, y) and the region R.
; R is the rectangle defined by and . Unlock Deck
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Unlock Deck
k this deck
51
Evaluate the double integral for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 2y + 3x; R is the rectangle defined by and .
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 2y + 3x; R is the rectangle defined by and . Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
52
Find the average value of the given function
Over the plane region R.
; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7.
A)
B)
C)
D)
Over the plane region R.
; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7.
A)
B)
C)
D)
Unlock Deck
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k this deck
53
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R.
; R is bounded by x = 0, x = 1, y = 0 and y = x. Unlock Deck
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k this deck
54
Evaluate the double integral for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x. Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
55
Use a double integral to find the volume of the solid shown in the figure.
Unlock Deck
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Unlock Deck
k this deck
56
The population density of a certain city is given by the function
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
57
Evaluate the double integral for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1. Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
58
Find the average value of the given function
F(x, y)
Over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
F(x, y)
Over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
A)
B)
C)
D)
Unlock Deck
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Unlock Deck
k this deck
59
Evaluate the double integral
for the function and the region R. Enter your answer as an expression.
and R is bounded by , , and .
for the function
and the region R. Enter your answer as an expression.
and R is bounded by , , and . Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
60
Evaluate the double integral
for the given function f(x, y) and the region R.
; R is the rectangle defined by and .
for the given function f(x, y) and the region R.
; R is the rectangle defined by and . Unlock Deck
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Unlock Deck
k this deck
61
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
62
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
63
Find the approximate change in z when the point changes from to . ; from to
A)
B)
C)
D)
E)
changes from to . ; from to A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
64
Find the average value of the given function
f(x, y)
over the plane region R. Enter your answer as a fraction.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).
f(x, y)
over the plane region R. Enter your answer as a fraction.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5). Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
65
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
66
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the region bounded by , and .
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the region bounded by , and . Unlock Deck
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Unlock Deck
k this deck
67
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).
z = f(x, y)
and below by the plane region R. Enter your answer as an expression.
; R is the triangle with vertices (0, 0), (4, 0) and (0, 4). Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
68
Use a double integral to find the volume of the solid shown in the figure. Enter your answer as an expression.
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
69
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
70
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where .
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where . Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
71
Find the approximate change in z when the point changes from to . ; from to
A)
B)
C)
D)
E)
changes from to . ; from to A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
72
Find the approximate change in z when the point changes from to . ; from to
A)
B)
C)
D)
E)
changes from to . ; from to A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
73
Evaluate the double integral
for the given function f(x, y) and the region R. Enter your answer as an expression.
; R is bounded by and y = x.
for the given function f(x, y) and the region R. Enter your answer as an expression.
; R is bounded by and y = x. Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
74
Find the average value of the given function
f(x, y)
over the plane region R. Enter your answer as an expression.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 7.
f(x, y)
over the plane region R. Enter your answer as an expression.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 7. Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
75
Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula.
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
76
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
77
Find the total differential of the function.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
78
Find the average value of the function
over the plane region R. Enter your answer as an expression.
and R is the triangle with vertices , and .
over the plane region R. Enter your answer as an expression.
and R is the triangle with vertices , and . Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
79
Find the volume of the solid bounded above by the surface
and below by the plane region R. Enter your answer as a formula.
and R is the region bounded by the graphs of and .
and below by the plane region R. Enter your answer as a formula.
and R is the region bounded by the graphs of and . Unlock Deck
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Unlock Deck
k this deck
80
The population density of a certain city is given by the function
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
. Enter your answer as an expression.
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
. Enter your answer as an expression. Unlock Deck
Unlock for access to all 219 flashcards in this deck.
Unlock Deck
k this deck
Unlock Deck
Unlock for access to all 219 flashcards in this deck.
