Question 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  .

Accepted Answer

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

Question 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

Accepted Answer

E

Question 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) ​

Accepted Answer

D

Question 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).

Accepted Answer

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

Question 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

Accepted Answer

The answer of Use a double integral to find the...

Question 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) ​

Accepted Answer

Using the washer method, the volume of the solid can be found by:
$$V=\int_{0}^{1} \pi\left[(2-x^2)^2-(x^2-1)^2ight]dx$$
Simplifying the integral and solving:
$$\begin{aligned} V&=\int_{0}^{1} \pi \left[4-5x^4+x^8ight]dx\ &=\pi\left[4x-\frac{5}{5}x^5+\frac{1}{9}x^9ight]_{0}^{1}\ &=\pi\left(4-\frac{5}{5}+\frac{1}{9}ight)\ &=\frac{28\pi}{9} \end{aligned}$$
Therefore, the answer is A.

Question 7

Use a double integral to find the volume of the solid shown in the figure.
​  ​
__________ cu units.

Accepted Answer

The answer of Use a double integral to find the...

Question 8

Use a double integral to find the volume of the solid shown in the figure. ​  ​
A) 
B) ​ 
C) ​ 
D) ​ 
E) ​

Accepted Answer

The answer of Use a double integral to find the...

Question 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  .

Accepted Answer

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

Question 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) ​

Accepted Answer

To find the population inside the rectangular area, we need to integrate the given function over the rectangular region.

The limits of integration for x are 0 to 1, and for y are 0 to 2.

∫[0,1] ∫[0,2] (1 + x + 2y) e^(-x-y) dy dx

= ∫[0,1] (1 + x + 4 - 4e^(-x)) e^(-x) dx

= [1 - 3e^(-x) + x - xe^(-x) + 4e^(-x)] [0,1]

= 11/2 - 2e^(-1)

≈ 2.27

Therefore, the population inside the rectangular area is approximately 2.27 million.

The closest answer choice is C) 2.3 million.

Question 11

Use a double integral to find the volume of the solid shown in the figure.  ​
A) 
B) ​ 
C) ​ 
D) ​ 
E) ​

Accepted Answer

The answer of Use a double integral to find the...

Question 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)

Accepted Answer

The answer of Find the average value of the function...

Question 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) ​

Accepted Answer

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

Question 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

Accepted Answer

The answer of Use a double integral to find the...

Question 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) ​

Accepted Answer

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

Question 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.

Accepted Answer

The answer of The Country Workshop's total weekly profit (in...

Question 17

Use a double integral to find the volume of the solid shown in the figure.
​  ​
__________ cu units.

Accepted Answer

The answer of Use a double integral to find the...

Question 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) ​

Accepted Answer

The answer of Find the average value of the given...

Question 19

Use a double integral to find the volume of the solid shown in the figure.
​

Accepted Answer

The answer of Use a double integral to find the...

Question 20

Use a double integral to find the volume of the solid shown in the figure.

Accepted Answer

The answer of Use a double integral to find the...

Quiz 8: Calculus of Several Variables

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

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) .

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).

Use a double integral to find the volume of the solid shown in the figure.

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 .

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

Use a double integral to find the volume of the solid shown in the figure.

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 .

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

Use a double integral to find the volume of the solid shown in the figure.

Find the average value of the function over the plane region R. and R is the triangle with vertices , and .

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 .

Use a double integral to find the volume of the solid shown in the figure.

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

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 .

Use a double integral to find the volume of the solid shown in the figure.

Use a double integral to find the volume of the solid shown in the figure.

Quiz 8: Calculus of Several Variables

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

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) . ​

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).

Use a double integral to find the volume of the solid shown in the figure. ​ ​

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 . ​

Use a double integral to find the volume of the solid shown in the figure. ​ ​ __________ cu units.

Use a double integral to find the volume of the solid shown in the figure. ​ ​

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 .

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 ​ ​

Use a double integral to find the volume of the solid shown in the figure. ​

Find the average value of the function over the plane region R. and R is the triangle with vertices , and . ​

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 . ​

Use a double integral to find the volume of the solid shown in the figure. ​ ​

Find the volume of the solid bounded above by the surface and below by the plane region R. ​ ; R is the region bounded by . ​

Use a double integral to find the volume of the solid shown in the figure. ​ ​ __________ cu units.

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 . ​

Use a double integral to find the volume of the solid shown in the figure. ​

Use a double integral to find the volume of the solid shown in the figure.

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) .

Use a double integral to find the volume of the solid shown in the figure.

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 .

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

Use a double integral to find the volume of the solid shown in the figure.

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 .

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

Use a double integral to find the volume of the solid shown in the figure.

Find the average value of the function over the plane region R. and R is the triangle with vertices , and .

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 .

Use a double integral to find the volume of the solid shown in the figure.

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

Use a double integral to find the volume of the solid shown in the figure. __________ cu units.

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 .

Use a double integral to find the volume of the solid shown in the figure.