Question 1

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 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 2

The solution of the initial value problem   = -   , y(0) = -3, is y = ±   .

Accepted Answer

A) True 
 B)False

Question 3

Solve the initial-value problem   - 2t(2x - 1) = 0, x(0) = 0.

Accepted Answer

A)  x = 1 -   
B)  x = -   +     
C)  x = -1 +   
D)  x =   -     
E)  x = 2 - 2   
A)  x = 1 -   
B)  x = -   +     
C)  x = -1 +   
D)  x =   -     
E)  x = 2 - 2

Question 4

A cubical die has each of the numbers 1 to 6 on one of its six faces. Unlike most such dice, however, this one is not at all fair. In fact, if X represents the number showing on top when the die is rolled, then   for 1 $\le$ j $\le$ 6, where C is a constant. Find the value of C and Pr(2 $\le$ X $\le$ 4).

Accepted Answer

A)  C =   , Pr(2 $\le$ X $\le$ 4) =   
B)  C =   , Pr(2 $\le$ X $\le$ 4) =   
C)  C =   , Pr(2 $\le$ X $\le$ 4) =   
D)  C =   , Pr(2 $\le$ X $\le$ 4) =   
E)  C =   , Pr(2 $\le$ X $\le$ 4) =   
A)  C =   , Pr(2 $\le$ X $\le$ 4) =   
B)  C =   , Pr(2 $\le$ X $\le$ 4) =   
C)  C =   , Pr(2 $\le$ X $\le$ 4) =   
D)  C =   , Pr(2 $\le$ X $\le$ 4) =   
E)  C =   , Pr(2 $\le$ X $\le$ 4) =

Question 5

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r). Find the volume of the remaining part of the ball.

Accepted Answer

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

Question 6

The solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis.

Accepted Answer

A) True 
 B)False

Question 7

Given the density function f(x) = 12   (1 - x) on [0, 1] and 0 elsewhere, find the variance and the standard deviation.

Accepted Answer

A)  mean   , standard deviation   
B)  mean   , standard deviation   
C)  mean   , standard deviation   
D)  mean   , standard deviation   
E)  mean 1, standard deviation   
A)  mean   , standard deviation   
B)  mean   , standard deviation   
C)  mean   , standard deviation   
D)  mean   , standard deviation   
E)  mean 1, standard deviation

Question 8

A force of 50 N is required to hold a spring that has been stretched from its natural length of 10 cm to 60 cm. How much work (in N . m) had to be done to stretch it that far?

Accepted Answer

A)  12.5 N . m 
B)  14.0 N . m 
C)  25.0 N . m 
D)  6.25 N . m 
E)  14.6 N . m 
A)  12.5 N . m 
B)  14.0 N . m 
C)  25.0 N . m 
D)  6.25 N . m 
E)  14.6 N . m

Question 9

A hemispherical bowl of radius r cm is filled with water. How far below the surface is the centre of mass of this water?

Accepted Answer

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

Question 10

Find the general solution of the differential equation   + tx = -2t.

Accepted Answer

A)  x = -1 + C   
B)  x = -2 + C   
C)  x = 2 + C   
D)  x = 1 + C   
E)  x = 2 + C   
A)  x = -1 + C   
B)  x = -2 + C   
C)  x = 2 + C   
D)  x = 1 + C   
E)  x = 2 + C

Question 11

A thin plate occupying the planar region 0 $\le$ x $\le$g(y), c $\le$ y $\le$ d has mass equal to 2 units. If the areal density   (y) =   ,   = 6, and   = 16, then the centre of mass of the plate is at the point:

Accepted Answer

A)  (   ,   ) = (16, 12) 
B)  (   ,   ) = (12, 16) 
C)  (   ,   ) = (4, 3) 
D)  (   ,   ) = (0, 0) 
E)  (   ,   ) = (3, 4) 
A)  (   ,   ) = (16, 12) 
B)  (   ,   ) = (12, 16) 
C)  (   ,   ) = (4, 3) 
D)  (   ,   ) = (0, 0) 
E)  (   ,   ) = (3, 4)

Question 12

If a school bus is travelling 90 km/hr on dry pavement, the stopping distance is approximately normal with a mean of 100 m and a standard deviation of 10 m. What is the probability that such a school bus could stop within a distance of 80 m?

Accepted Answer

A)  0.014 
B)  0.075 
C)  0.023 
D)  0.025 
E)  0.077 
A)  0.014 
B)  0.075 
C)  0.023 
D)  0.025 
E)  0.077

Question 13

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 y-axis.

Accepted Answer

A)  2   cubic units 
B)    cubic units 
C)  3   cubic units 
D)    cubic units 
E)  4   cubic units 
A)  2   cubic units 
B)    cubic units 
C)  3   cubic units 
D)    cubic units 
E)  4   cubic units

Question 14

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve   above the x-axis and to the left of the y-axis.

Accepted Answer

A)  32$\pi$ cubic units 
B)  16$\pi$ cubic units 
C)  64$\pi$ cubic units 
D)  4$\pi$ cubic units 
E)  25$\pi$ cubic units 
A)  32$\pi$ cubic units 
B)  16$\pi$ cubic units 
C)  64$\pi$ cubic units 
D)  4$\pi$ cubic units 
E)  25$\pi$ cubic units

Question 15

A normal distribution has a mean of 55, and about 25% of the values are above 65. Approximately what is the standard deviation for this distribution?

Accepted Answer

A)  15 
B)  18 
C)  13 
D)  20 
E)  10 
A)  15 
B)  18 
C)  13 
D)  20 
E)  10

Question 16

Solve the initial-value problem   + 5y = 10, y(0) = 10.

Accepted Answer

A)  y(x) =   + 9 
B)  y(x) = 8   + 2 
C)  y(x) = 5   + 5 
D)  y(x) = 8   + 2 
E)  y(x) = 8   + 2 
A)  y(x) =   + 9 
B)  y(x) = 8   + 2 
C)  y(x) = 5   + 5 
D)  y(x) = 8   + 2 
E)  y(x) = 8   + 2

Question 17

Find the centroid of the finite plane region bounded by the curve y = 4 - x2 and the liney = x + 2.

Accepted Answer

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

Question 18

Find the area of the surface obtained by rotating the curve y =   , -1 $\le$ x $\le$ 1, about the y-axis.

Accepted Answer

A)    (5   - 1) square units 
B)    (5   + 1) square units 
C)    (5   - 1) square units 
D)    (5   + 1) square units 
E)    (5   - 1) square units 
A)    (5   - 1) square units 
B)    (5   + 1) square units 
C)    (5   - 1) square units 
D)    (5   + 1) square units 
E)    (5   - 1) square units

Question 19

Find the volume of the solid generated by revolving the region enclosed by the graphs of y =   and the x-axis from x = 0 to x = 1 about the y-axis.

Accepted Answer

A)  2$\pi$ (2e -1) 
B)  2$\pi$ 
C)  $\pi$ e 
D)    
E)  $\pi$   
A)  2$\pi$ (2e -1) 
B)  2$\pi$ 
C)  $\pi$ e 
D)    
E)  $\pi$

Question 20

Find the moment about the line x = 4 of a plate of constant density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - x2.

Accepted Answer

A)  32 
B)  36 
C)  34 
D)  38 
E)  30 
A)  32 
B)  36 
C)  34 
D)  38 
E)  30

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 x-axis.

The solution of the initial value problem = - , y(0) = -3, is y = ± .

Solve the initial-value problem - 2t(2x - 1) = 0, x(0) = 0.

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r). Find the volume of the remaining part of the ball.

The solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis.

Given the density function f(x) = 12 (1 - x) on [0, 1] and 0 elsewhere, find the variance and the standard deviation.

A force of 50 N is required to hold a spring that has been stretched from its natural length of 10 cm to 60 cm. How much work (in N . m) had to be done to stretch it that far?

A hemispherical bowl of radius r cm is filled with water. How far below the surface is the centre of mass of this water?

Find the general solution of the differential equation + tx = -2t.

If a school bus is travelling 90 km/hr on dry pavement, the stopping distance is approximately normal with a mean of 100 m and a standard deviation of 10 m. What is the probability that such a school bus could stop within a distance of 80 m?

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 y-axis.

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve above the x-axis and to the left of the y-axis.

A normal distribution has a mean of 55, and about 25% of the values are above 65. Approximately what is the standard deviation for this distribution?

Solve the initial-value problem + 5y = 10, y(0) = 10.

Find the centroid of the finite plane region bounded by the curve y = 4 - x² and the liney = x + 2.

Find the area of the surface obtained by rotating the curve y = $Find the area of the surface obtained by rotating the curve y = , -1 \le x \le 1, about the y-axis.$ , -1 $\le$ x $\le$ 1, about the y-axis.

Find the volume of the solid generated by revolving the region enclosed by the graphs of y = and the x-axis from x = 0 to x = 1 about the y-axis.

Find the moment about the line x = 4 of a plate of constant density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - x².

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

The solution of the initial value problem = - , y(0) = -3, is y = ± .

Solve the initial-value problem - 2t(2x - 1) = 0, x(0) = 0.

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r). Find the volume of the remaining part of the ball.

The solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis.

Given the density function f(x) = 12 (1 - x) on [0, 1] and 0 elsewhere, find the variance and the standard deviation.

A force of 50 N is required to hold a spring that has been stretched from its natural length of 10 cm to 60 cm. How much work (in N . m) had to be done to stretch it that far?

A hemispherical bowl of radius r cm is filled with water. How far below the surface is the centre of mass of this water?

Find the general solution of the differential equation + tx = -2t.

A thin plate occupying the planar region 0 ≤\le≤ x ≤\le≤ g(y), c ≤\le≤ y ≤\le≤ d has mass equal to 2 units. If the areal density (y) = , = 6, and = 16, then the centre of mass of the plate is at the point:

If a school bus is travelling 90 km/hr on dry pavement, the stopping distance is approximately normal with a mean of 100 m and a standard deviation of 10 m. What is the probability that such a school bus could stop within a distance of 80 m?

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 y-axis.

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve above the x-axis and to the left of the y-axis.

A normal distribution has a mean of 55, and about 25% of the values are above 65. Approximately what is the standard deviation for this distribution?

Solve the initial-value problem + 5y = 10, y(0) = 10.

Find the centroid of the finite plane region bounded by the curve y = 4 - x2 and the liney = x + 2.

Find the area of the surface obtained by rotating the curve y = , -1 ≤\le≤ x ≤\le≤ 1, about the y-axis.

Find the volume of the solid generated by revolving the region enclosed by the graphs of y = and the x-axis from x = 0 to x = 1 about the y-axis.

Find the moment about the line x = 4 of a plate of constant density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - x2.

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

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 y-axis.

Find the centroid of the finite plane region bounded by the curve y = 4 - x² and the liney = x + 2.

Find the area of the surface obtained by rotating the curve y = $Find the area of the surface obtained by rotating the curve y = , -1 \le x \le 1, about the y-axis.$ , -1 $\le$ x $\le$ 1, about the y-axis.

Find the moment about the line x = 4 of a plate of constant density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - x².