Question 1

Determine the centroid of the finite plane region bounded by y = x3 and y =   .

Accepted Answer

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

Question 2

Find the area of the surface generated by rotating y =   , -$\infty$ $\le$ x $\le$ 0 about y = 0.

Accepted Answer

A)  $\pi$   square units 
B)  $\pi$   square units 
C)  $\pi$   square units 
D)    + ln(1 +   ) square units 
E)    - ln(1 +   ) square units 
A)  $\pi$   square units 
B)  $\pi$   square units 
C)  $\pi$   square units 
D)    + ln(1 +   ) square units 
E)    - ln(1 +   ) square units

Question 3

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting 10 years from now? Assume an interest rate of 8% per annum, compounded continuously. Round to the nearest dollar.

Accepted Answer

A)  $56,166 
B)  $54,892 
C)  $61,720 
D)  $59,010 
E)  $68,834 
A)  $56,166 
B)  $54,892 
C)  $61,720 
D)  $59,010 
E)  $68,834

Question 4

A triangle T has vertices (   ,   ), j = 1, 2, 3 (where each   > 0). If the volume of the solid of revolution obtained by revolving T about the y-axis is 2$\pi$ cubic units, what is the area of T?

Accepted Answer

A)    square units 
B)    square units 
C)    square units 
D)    square units 
E)    square units 
A)    square units 
B)    square units 
C)    square units 
D)    square units 
E)    square units

Question 5

A pyramid has a triangular base of area A and has a height of h measured perpendicular to the plane of the base. Determine the volume of the pyramid.

Accepted Answer

A)    Ah cubic units 
B)  Ah cubic units 
C)    Ah cubic units 
D)    Ah cubic units 
E)  Ah$\pi$ cubic units 
A)    Ah cubic units 
B)  Ah cubic units 
C)    Ah cubic units 
D)    Ah cubic units 
E)  Ah$\pi$ cubic units

Question 6

Determine the centre of mass for the region bounded by y = 2 sin 2x, y = 0 on the interval [0,   ].

Accepted Answer

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

Question 7

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated 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 8

Find the volume of the solid generated when the region lying under the curve y = 4 - x2 and above the x-axis is rotated about the line y = -1.

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

Find the total length of the hypocycloid   +   =   .

Accepted Answer

A)  12a units 
B)  10a units 
C)  8a units 
D)  6a units 
E)  $\pi$a units 
A)  12a units 
B)  10a units 
C)  8a units 
D)  6a units 
E)  $\pi$a units

Question 10

The equations y = x3, y = 0, and x = 1 define the bounds of a plane region. 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 11

Find the centroid of the plane quadrilateral region with corners at (0, 0), (0, 2), (2, 3), and (1, 0).

Accepted Answer

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

Question 12

The equations y2 = 4x and x2 = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y2 = 4x and above the curve x2 = 4y about
(a) the x-axis and
(b) the y-axis.

Accepted Answer

A)  (a)   cubic units, (b)   cubic units 
B)  (a)   cubic units, (b)   cubic units 
C)  (a)   cubic units, (b)   cubic units 
D)  (a)   cubic units, (b)   cubic units 
E)  (a)   cubic units, (b)   cubic units 
A)  (a)   cubic units, (b)   cubic units 
B)  (a)   cubic units, (b)   cubic units 
C)  (a)   cubic units, (b)   cubic units 
D)  (a)   cubic units, (b)   cubic units 
E)  (a)   cubic units, (b)   cubic units

Question 13

A normal distribution has a standard deviation of 25, and about 2.3% values are below 10. Find the mean for this distribution.

Accepted Answer

A)  60 
B)  70 
C)  40 
D)  50 
E)  10 
A)  60 
B)  70 
C)  40 
D)  50 
E)  10

Question 14

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting now? Assume an interest rate of 8% per annum, compounded continuously.

Accepted Answer

A)  $135,000 
B)  $130,000 
C)  $125,000 
D)  $120,000 
E)  $115,000 
A)  $135,000 
B)  $130,000 
C)  $125,000 
D)  $120,000 
E)  $115,000

Question 15

A triangular plate has vertices at (0, 0), (a, 0), and (0,b), where a > 0 and b > 0. The plate has variable thickness; at position (x, y) its thickness is   . Assuming the plate is made of material of constant density, find the x-coordinate of its centre of mass.

Accepted Answer

The answer of A triangular plate has vertices at (0,...

Question 16

Solve the initial-value problem   = (t + 1)y, y(0) = 3.

Accepted Answer

A)  y(x) = 3exp   
B)  y(x) = 3exp   
C)  y(x) = 3exp   
D)  y(x) = 3exp   
E)  y(x) = 3exp   
A)  y(x) = 3exp   
B)  y(x) = 3exp   
C)  y(x) = 3exp   
D)  y(x) = 3exp   
E)  y(x) = 3exp

Question 17

Hooke's law states that the force required to stretch or compress a spring to x units longer or shorter than its natural length is, for sufficiently small values of x, proportional to x. That is, F(x) = kx for some constant k, called the spring constant. Suppose that a force of 10 N is required to stretch a spring by 5 cm. Find the work done in stretching the spring that far.

Accepted Answer

A)  25 N . cm 
B)  22 N . cm 
C)  27 N . cm 
D)  29 N . cm 
E)  20 N . cm 
A)  25 N . cm 
B)  22 N . cm 
C)  27 N . cm 
D)  29 N . cm 
E)  20 N . cm

Question 18

Find the centroid of the region in the first quadrant bounded by the lines y = 5x, y = x, and x = 4.

Accepted Answer

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

Question 19

The plane region defined by 0 $\le$ y $\le$   , 0 $\le$ x $\le$ 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

A)  on the x-axis,   units to the right of the origin 
B)  on the x-axis,   units to the right of the origin 
C)  on the x-axis,   units to the right of the origin 
D)  on the x-axis,   units to the right of the origin 
E)  on the x-axis,   units to the right of the origin 
A)  on the x-axis,   units to the right of the origin 
B)  on the x-axis,   units to the right of the origin 
C)  on the x-axis,   units to the right of the origin 
D)  on the x-axis,   units to the right of the origin 
E)  on the x-axis,   units to the right of the origin

Question 20

Find the moment about the x-axis of the region in the first quadrant bounded by the lines y = 5x, y = 3x, x = 3. Assume the areal density is 1.

Accepted Answer

A)  16 
B)  18 
C)  17 
D)  20 
E)  14 
A)  16 
B)  18 
C)  17 
D)  20 
E)  14

Determine the centroid of the finite plane region bounded by y = x³ and y = .

Find the area of the surface generated by rotating y = $Find the area of the surface generated by rotating y = , - \infty \le x \le 0 about y = 0.$ , - $\infty$ $\le$ x $\le$ 0 about y = 0.

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting 10 years from now? Assume an interest rate of 8% per annum, compounded continuously. Round to the nearest dollar.

A pyramid has a triangular base of area A and has a height of h measured perpendicular to the plane of the base. Determine the volume of the pyramid.

Determine the centre of mass for the region bounded by y = 2 sin 2x, y = 0 on the interval [0, ].

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.

Find the total length of the hypocycloid + = .

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

Find the centroid of the plane quadrilateral region with corners at (0, 0), (0, 2), (2, 3), and (1, 0).

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about (a) the x-axis and (b) the y-axis.

A normal distribution has a standard deviation of 25, and about 2.3% values are below 10. Find the mean for this distribution.

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting now? Assume an interest rate of 8% per annum, compounded continuously.

A triangular plate has vertices at (0, 0), (a, 0), and (0,b), where a > 0 and b > 0. The plate has variable thickness; at position (x, y) its thickness is . Assuming the plate is made of material of constant density, find the x-coordinate of its centre of mass.

Solve the initial-value problem = (t + 1)y, y(0) = 3.

Find the centroid of the region in the first quadrant bounded by the lines y = 5x, y = x, and x = 4.

Find the moment about the x-axis of the region in the first quadrant bounded by the lines y = 5x, y = 3x, x = 3. Assume the areal density is 1.

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

Determine the centroid of the finite plane region bounded by y = x3 and y = .

Find the area of the surface generated by rotating y = , - ∞\infty∞ ≤\le≤ x ≤\le≤ 0 about y = 0.

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting 10 years from now? Assume an interest rate of 8% per annum, compounded continuously. Round to the nearest dollar.

A triangle T has vertices ( , ), j = 1, 2, 3 (where each > 0). If the volume of the solid of revolution obtained by revolving T about the y-axis is 2 π\piπ cubic units, what is the area of T?

A pyramid has a triangular base of area A and has a height of h measured perpendicular to the plane of the base. Determine the volume of the pyramid.

Determine the centre of mass for the region bounded by y = 2 sin 2x, y = 0 on the interval [0, ].

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = π\piπ is rotated about the x-axis.

Find the volume of the solid generated when the region lying under the curve y = 4 - x2 and above the x-axis is rotated about the line y = -1.

Find the total length of the hypocycloid + = .

The equations y = x3, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

Find the centroid of the plane quadrilateral region with corners at (0, 0), (0, 2), (2, 3), and (1, 0).

The equations y2 = 4x and x2 = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y2 = 4x and above the curve x2 = 4y about (a) the x-axis and (b) the y-axis.

A normal distribution has a standard deviation of 25, and about 2.3% values are below 10. Find the mean for this distribution.

What is the present value of a constant, continual stream of payments at a rate of $10,000 per year to continue forever starting now? Assume an interest rate of 8% per annum, compounded continuously.

A triangular plate has vertices at (0, 0), (a, 0), and (0,b), where a > 0 and b > 0. The plate has variable thickness; at position (x, y) its thickness is . Assuming the plate is made of material of constant density, find the x-coordinate of its centre of mass.

Solve the initial-value problem = (t + 1)y, y(0) = 3.

Find the centroid of the region in the first quadrant bounded by the lines y = 5x, y = x, and x = 4.

The plane region defined by 0 ≤\le≤ y ≤\le≤ , 0 ≤\le≤ x ≤\le≤ 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 moment about the x-axis of the region in the first quadrant bounded by the lines y = 5x, y = 3x, x = 3. Assume the areal density is 1.

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

Determine the centroid of the finite plane region bounded by y = x³ and y = .

Find the area of the surface generated by rotating y = $Find the area of the surface generated by rotating y = , - \infty \le x \le 0 about y = 0.$ , - $\infty$ $\le$ x $\le$ 0 about y = 0.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about (a) the x-axis and (b) the y-axis.