Question 1

Find the value of the constant real number C so that f(x) = C(x -   ) is a probability density function on the interval [0 , 1].

Accepted Answer

A)    
B)  1 
C)  6 
D)  - 6 
E)  -1 
A)    
B)  1 
C)  6 
D)  - 6 
E)  -1

Question 2

The plane region defined by 0 ≤ y ≤   , 0 ≤ x ≤ a is revolved about the x-axis to generate a 3-dimensional region that is filled with material of constant density. Where is the centre of mass of this material?

Accepted Answer

On the x-a

Question 3

The random variable T has density function f(t), where f(t) =     if t $\ge$ 0 and    is a positive constant. Find the standard deviation of T.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 4

Find the centroid of the region bounded by the x-axis and the curve y = -16 + 10 x - x2.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

A supplier of cell phones realizes a marginal revenue of 25 - 15   dollars per cell phone when he has sold x cell phones. What will be his total revenue from the sale of 100 cell phones?

Accepted Answer

A)  $1551.82 
B)  $1400.34 
C)  $1666.66 
D)  $1677.66 
E)  $1504.34 
A)  $1551.82 
B)  $1400.34 
C)  $1666.66 
D)  $1677.66 
E)  $1504.34

Question 6

Find the solution of the initial-value problem   = 3   , y(0) = 0.

Accepted Answer

A)  y =     - 12 
B)  y = 9   - 36 
C)  y =     -   
D)  y = -     + 12 
E)  y = 3   - 48 
A)  y =     - 12 
B)  y = 9   - 36 
C)  y =     -   
D)  y = -     + 12 
E)  y = 3   - 48

Question 7

A conical reservoir (vertex down) having diameter 10 m and height 10 m is filled with a liquid of density 900 kg/m3. How many kilogram metres of work must be done to pump the liquid out over the top of the tank?

Accepted Answer

A)  187,500$\pi$ kg . m 
B)  167,500$\pi$ kg . m 
C)  175,000$\pi$ kg . m 
D)  204,000$\pi$ kg . m 
E)  1,837,000$\pi$ kg . m 
A)  187,500$\pi$ kg . m 
B)  167,500$\pi$ kg . m 
C)  175,000$\pi$ kg . m 
D)  204,000$\pi$ kg . m 
E)  1,837,000$\pi$ kg . m

Question 8

The equations x = -1, x = 0, y =   , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Accepted Answer

A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units 
A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units

Question 9

For what real values of the constant k does the region lying under the curve y =   above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

Accepted Answer

A)    < k$\le$ 1 
B)    < k < 1 
C)    < k < $\infty$ 
D)  0 < k <   
E)  0 < k $\le$ 1 
A)    < k$\le$ 1 
B)    < k < 1 
C)    < k < $\infty$ 
D)  0 < k <   
E)  0 < k $\le$ 1

Question 10

Find the volume of the solid obtained by rotating the region inside the circle x2 + y2 = 6 and above the parabola y = x2 about the y-axis.

Accepted Answer

A)  8$\pi$   -   cubic units 
B)  4$\pi$   -   cubic units 
C)  4$\pi$   -   cubic units 
D)  8$\pi$   -   cubic units 
E)  2$\pi$  -   cubic units 
A)  8$\pi$   -   cubic units 
B)  4$\pi$   -   cubic units 
C)  4$\pi$   -   cubic units 
D)  8$\pi$   -   cubic units 
E)  2$\pi$  -   cubic units

Question 11

Find the centre of mass of a system of point masses m1 = 6, m2 = 3, m3 = 2, and m4 = 9 located at (3, -2), (0, 0), (-5, 3), and (4, 2), respectively.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line x = -2 is V = 2$\pi$    dx.

Accepted Answer

A) True 
 B)False

Question 13

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x2 - 2x.

Accepted Answer

A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units 
A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units

Question 14

A force of f pounds acting along the x-axis varies according to f(x) = x2 + 3x - 2. Find the work it does on an object that moves from x = -2 to x = 2.

Accepted Answer

A)  -   
B)  -   
C)  -   
D)  -   
E)  -   
A)  -   
B)  -   
C)  -   
D)  -   
E)  -

Question 15

Find a general solution of the first-order linear differential equation   + 3y = e6x.

Accepted Answer

A)  y =     +   
B)  y =     + C   
C)  y = -     + C   
D)  y =     + C   
E)  y =     + C   
A)  y =     +   
B)  y =     + C   
C)  y = -     + C   
D)  y =     + C   
E)  y =     + C

Question 16

Let X have density function f(x), where f(x) =   when x $\ge$ 1. Find the mean and standard deviation of X.

Accepted Answer

A)    = 2,   = 3 
B)    = 2,  = 2 
C)    = 1,   = 3 
D)    = 2,   = does not exist 
E)  neither the mean nor the standard deviation exist 
A)    = 2,   = 3 
B)    = 2,  = 2 
C)    = 1,   = 3 
D)    = 2,   = does not exist 
E)  neither the mean nor the standard deviation exist

Question 17

Find the centroid of the finite plane region bounded by y = x2 and y = x.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

Find an integrating factor for the differential equation   + y = x.

Accepted Answer

A)    (x) =   
B)   (x) =   
C)    (x) =   
D)    (x) =   
E)    (x) =   
A)    (x) =   
B)   (x) =   
C)    (x) =   
D)    (x) =   
E)    (x) =

Question 19

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Accepted Answer

A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units 
A)    cubic units 
B)    cubic units 
C)    cubic units 
D)    cubic units 
E)    cubic units

Question 20

Find the general solution of the differential equation       =

Accepted Answer

A)  x = -2   
B)  x = 2   
C)  x = -   
D)  x =   
E)  x =  + C 
A)  x = -2   
B)  x = 2   
C)  x = -   
D)  x =   
E)  x =  + C

Find the value of the constant real number C so that f(x) = C(x - ) is a probability density function on the interval [0 , 1].

The plane region defined by 0 ≤ y ≤ , 0 ≤ x ≤ a is revolved about the x-axis to generate a 3-dimensional region that is filled with material of constant density. Where is the centre of mass of this material?

Find the centroid of the region bounded by the x-axis and the curve y = -16 + 10 x - x².

A supplier of cell phones realizes a marginal revenue of 25 - 15 dollars per cell phone when he has sold x cell phones. What will be his total revenue from the sale of 100 cell phones?

Find the solution of the initial-value problem = 3 , y(0) = 0.

A conical reservoir (vertex down) having diameter 10 m and height 10 m is filled with a liquid of density 900 kg/m³. How many kilogram metres of work must be done to pump the liquid out over the top of the tank?

The equations x = -1, x = 0, y = , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

For what real values of the constant k does the region lying under the curve y = above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

Find the centre of mass of a system of point masses m₁ = 6, m₂ = 3, m₃ = 2, and m₄ = 9 located at (3, -2), (0, 0), (-5, 3), and (4, 2), respectively.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

A force of f pounds acting along the x-axis varies according to f(x) = x² + 3x - 2. Find the work it does on an object that moves from x = -2 to x = 2.

Find a general solution of the first-order linear differential equation + 3y = e^6x.

Let X have density function f(x), where f(x) = $Let X have density function f(x), where f(x) = when x \ge 1. Find the mean and standard deviation of X.$ when x $\ge$ 1. Find the mean and standard deviation of X.

Find the centroid of the finite plane region bounded by y = x² and y = x.

Find an integrating factor for the differential equation + y = x.

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Find the general solution of the differential equation =

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 8: Applications of Integration

Find the value of the constant real number C so that f(x) = C(x - ) is a probability density function on the interval [0 , 1].

The plane region defined by 0 ≤ y ≤ , 0 ≤ x ≤ a is revolved about the x-axis to generate a 3-dimensional region that is filled with material of constant density. Where is the centre of mass of this material?

The random variable T has density function f(t), where f(t) = if t ≥\ge≥ 0 and 11ee7b36_e41e_108a_ae82_6b99642b54b5_TB9661_11 is a positive constant. Find the standard deviation of T.

Find the centroid of the region bounded by the x-axis and the curve y = -16 + 10 x - x2.

A supplier of cell phones realizes a marginal revenue of 25 - 15 dollars per cell phone when he has sold x cell phones. What will be his total revenue from the sale of 100 cell phones?

Find the solution of the initial-value problem = 3 , y(0) = 0.

A conical reservoir (vertex down) having diameter 10 m and height 10 m is filled with a liquid of density 900 kg/m3. How many kilogram metres of work must be done to pump the liquid out over the top of the tank?

The equations x = -1, x = 0, y = , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

For what real values of the constant k does the region lying under the curve y = above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

Find the volume of the solid obtained by rotating the region inside the circle x2 + y2 = 6 and above the parabola y = x2 about the y-axis.

Find the centre of mass of a system of point masses m1 = 6, m2 = 3, m3 = 2, and m4 = 9 located at (3, -2), (0, 0), (-5, 3), and (4, 2), respectively.

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line x = -2 is V = 2 π\piπ dx.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x2 - 2x.

A force of f pounds acting along the x-axis varies according to f(x) = x2 + 3x - 2. Find the work it does on an object that moves from x = -2 to x = 2.

Find a general solution of the first-order linear differential equation + 3y = e6x.

Let X have density function f(x), where f(x) = when x ≥\ge≥ 1. Find the mean and standard deviation of X.

Find the centroid of the finite plane region bounded by y = x2 and y = x.

Find an integrating factor for the differential equation + y = x.

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Find the general solution of the differential equation =

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find the centroid of the region bounded by the x-axis and the curve y = -16 + 10 x - x².

A conical reservoir (vertex down) having diameter 10 m and height 10 m is filled with a liquid of density 900 kg/m³. How many kilogram metres of work must be done to pump the liquid out over the top of the tank?

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

Find the centre of mass of a system of point masses m₁ = 6, m₂ = 3, m₃ = 2, and m₄ = 9 located at (3, -2), (0, 0), (-5, 3), and (4, 2), respectively.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

A force of f pounds acting along the x-axis varies according to f(x) = x² + 3x - 2. Find the work it does on an object that moves from x = -2 to x = 2.

Find a general solution of the first-order linear differential equation + 3y = e^6x.

Let X have density function f(x), where f(x) = $Let X have density function f(x), where f(x) = when x \ge 1. Find the mean and standard deviation of X.$ when x $\ge$ 1. Find the mean and standard deviation of X.

Find the centroid of the finite plane region bounded by y = x² and y = x.