Question 1

Find the center of mass of a thin plate covering the given region with the given density function.
-The region bounded below by the parabola  and above by the line  , with density

Accepted Answer

A)  
B)  
C)  
D)

Question 2

Write an equivalent double integral with the order of integration reversed.
-

Accepted Answer

A)   
B)   
C)   
D)

Question 3

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by y = 6 - x and the axes

Accepted Answer

A)  
B)  
C)  
D)

Question 4

Write an equivalent double integral with the order of integration reversed.
-

Accepted Answer

A)   
B)   
C)   
D)

Question 5

Solve the problem.
-Find the center of mass of the region of constant density bounded by the paraboloid  and the xy-plane.

Accepted Answer

A)  
B)  
C)  
D)

Question 6

Use the given transformation to evaluate the integral.
- where R is the parallelogram bounded by the lines

Accepted Answer

A)  756 
B)   
C)   
D)  378

Question 7

Find the Jacobian for the given transformation
-

Accepted Answer

A)  120  
B)  60  
C)  60  
D)  120

Question 8

Solve the problem.
-Let D be the smaller cap cut from a solid ball of radius 7 units by a plane 4 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Accepted Answer

A)   
B)   
C)   
D)

Question 9

Evaluate the integral by changing the order of integration in an appropriate way.
-

Accepted Answer

A)   ln 2 (e - 1) 
B)  2 ln 3 (e - 1) 
C)  2 ln 2 (e - 1) 
D)   ln 3 (e - 1)

Question 10

Evaluate the spherical coordinate integral.
-

Accepted Answer

A)  12$\pi$ 
B)  12 
C)  18 
D)  18$\pi$

Question 11

Solve the problem.
-If f(x, y) = ( 3000  )/(1 +  /2) represents the population density of a planar region on Earth, where x and y are measured in miles, find the number of people within the rectangle -7 $\le$ x $\le$ 7 and -3 $\le$ y $\le$ 0.

Accepted Answer

A)   
B)  
C)  
D)

Question 12

Find the volume of the indicated region.
-the region bounded by the paraboloid  and the cylinder

Accepted Answer

A)  $\pi$ 
B)   $\pi$ 
C)   $\pi$ 
D)  $\pi$

Question 13

Use the given transformation to evaluate the integral.
- where R is the interior of the ellipsoid

Accepted Answer

A)   
B)   
C)   
D)

Question 14

Evaluate the integral.
-

Accepted Answer

A)  5 
B)   
C)  37 
D)

Question 15

Evaluate the double integral over the given region.
-  R = {(x, y):  9$\le$x$\le$10, 1$\le$y$\le$5}

Accepted Answer

A)  114 
B)  152 
C)  228 
D)  76

Question 16

Find the area of the region specified in polar coordinates.
-the region inside  and outside

Accepted Answer

A)  100  
B)  100  
C)  100  
D)  100

Question 17

Integrate the function f over the given region.
-f(x, y) =  +  over the trapezoidal region bounded by the x-axis, y-axis, line  and line

Accepted Answer

A)   
B)   
C)   
D)

Question 18

Find the center of mass of a thin plate of constant density covering the given region.
-The region between the curve y =  and the x-axis from  to

Accepted Answer

A)  
B)  
C)  
D)

Question 19

Integrate the function f over the given region.
-f(x, y) =  over the region bounded by the x-axis, line  and curve

Accepted Answer

A)  7 
B)  1 
C)  8 
D)  9

Exam 16: Multiple Integration

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded below by the parabola and above by the line , with density

Write an equivalent double integral with the order of integration reversed. -

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by y = 6 - x and the axes

Write an equivalent double integral with the order of integration reversed. -

Solve the problem. -Find the center of mass of the region of constant density bounded by the paraboloid and the xy-plane.

Use the given transformation to evaluate the integral. - where R is the parallelogram bounded by the lines

Find the Jacobian for the given transformation -

Solve the problem. -Let D be the smaller cap cut from a solid ball of radius 7 units by a plane 4 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Evaluate the integral by changing the order of integration in an appropriate way. -

Evaluate the spherical coordinate integral. - $Evaluate the spherical coordinate integral. - A) 12 \pi B) 12 C) 18 D) 18 \pi$

Use the given transformation to evaluate the integral. - where R is the interior of the ellipsoid

Evaluate the integral. -

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y) : 9 \le x \le 10, 1 \le y \le 5} A) 114 B) 152 C) 228 D) 76$ R = {(x, y) : 9 $\le$ x $\le$ 10, 1 $\le$ y $\le$ 5}

Find the area of the region specified in polar coordinates. -the region inside and outside

Integrate the function f over the given region. -f(x, y) = + over the trapezoidal region bounded by the x-axis, y-axis, line and line

Find the center of mass of a thin plate of constant density covering the given region. -The region between the curve y = and the x-axis from to

Integrate the function f over the given region. -f(x, y) = over the region bounded by the x-axis, line and curve

Exam 16: Multiple Integration

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded below by the parabola and above by the line , with density

Write an equivalent double integral with the order of integration reversed. -

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by y = 6 - x and the axes

Write an equivalent double integral with the order of integration reversed. -

Solve the problem. -Find the center of mass of the region of constant density bounded by the paraboloid and the xy-plane.

Use the given transformation to evaluate the integral. - where R is the parallelogram bounded by the lines

Find the Jacobian for the given transformation -

Solve the problem. -Let D be the smaller cap cut from a solid ball of radius 7 units by a plane 4 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Evaluate the integral by changing the order of integration in an appropriate way. -

Evaluate the spherical coordinate integral. -

Solve the problem. -If f(x, y) = ( 3000 ) /(1 + /2) represents the population density of a planar region on Earth, where x and y are measured in miles, find the number of people within the rectangle -7 ≤\le≤ x ≤\le≤ 7 and -3 ≤\le≤ y ≤\le≤ 0.

Find the volume of the indicated region. -the region bounded by the paraboloid and the cylinder

Use the given transformation to evaluate the integral. - where R is the interior of the ellipsoid

Evaluate the integral. -

Evaluate the double integral over the given region. - R = {(x, y) : 9 ≤\le≤ x ≤\le≤ 10, 1 ≤\le≤ y ≤\le≤ 5}

Find the area of the region specified in polar coordinates. -the region inside and outside

Integrate the function f over the given region. -f(x, y) = + over the trapezoidal region bounded by the x-axis, y-axis, line and line

Find the center of mass of a thin plate of constant density covering the given region. -The region between the curve y = and the x-axis from to

Integrate the function f over the given region. -f(x, y) = over the region bounded by the x-axis, line and curve

Evaluate the spherical coordinate integral. - $Evaluate the spherical coordinate integral. - A) 12 \pi B) 12 C) 18 D) 18 \pi$

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y) : 9 \le x \le 10, 1 \le y \le 5} A) 114 B) 152 C) 228 D) 76$ R = {(x, y) : 9 $\le$ x $\le$ 10, 1 $\le$ y $\le$ 5}