Question 1

A random variable X is normally distributed with mean 100 and standard deviation 20. Below what value of X does 25% of the probability lie?

Accepted Answer

A)  83 
B)  85 
C)  87 
D)  89 
E)  90 
A)  83 
B)  85 
C)  87 
D)  89 
E)  90

Question 2

For what constant k if f(x) = k   a probability density function on [0, 1]?

Accepted Answer

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

Question 3

Interior to the Earth, the gravitational attraction varies directly as the distance from the centre of the Earth. What is the work necessary to lift a mass weighing 1 kg at the surface from the centre to the surface? Assume that the radius of the Earth is 6,400 km and the force of attraction of the Earth on a   mass at its surface is 9.8 Newtons.

Accepted Answer

A)  24,260 N . km 
B)  31,360 N . km 
C)  36,480 N . km 
D)  42,400 N . km 
E)  none of the above 
A)  24,260 N . km 
B)  31,360 N . km 
C)  36,480 N . km 
D)  42,400 N . km 
E)  none of the above

Question 4

Find, correct to 4 decimal places, the length of the curve y =   from x = 1 to x = 8.

Accepted Answer

A)  19.1981 units 
B)  14.6572 units 
C)  3.4123 units 
D)  7.6337 units 
E)  22.8030 units 
A)  19.1981 units 
B)  14.6572 units 
C)  3.4123 units 
D)  7.6337 units 
E)  22.8030 units

Question 5

Find the volumes of solids generated when the ellipse   +   = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

Accepted Answer

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

Question 6

The base of a certain solid is a circular disk of radius a cm. Cross-sections of the solid in planes perpendicular to a specific diameter of the base are equilateral triangles. Find the volume of the solid.

Accepted Answer

A)        
B)        
C)  2       
D)        
E)  4       
A)        
B)        
C)  2       
D)        
E)  4

Question 7

Find the solution of the initial value problem   =   , y(π/2) = -   .

Accepted Answer

The answer of Find the solution of the initial value...

Question 8

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

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 9

Suppose the thickness of the blacktop on a road that has recently been resurfaced is normal with a mean of 6 cm and a standard deviation of 0.4 cm. What percentage of the blacktop is at least 5 cm thick?

Accepted Answer

A)  0.883 
B)  0.994 
C)  0.006 
D)  0.722 
E)  0.432 
A)  0.883 
B)  0.994 
C)  0.006 
D)  0.722 
E)  0.432

Question 10

Find the family of curves each of which intersects all of the hyperbolas x2 - y2 = C at right angles.

Accepted Answer

A family of (rectang

Question 11

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

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

Question 12

Plutonium 239 decays continuously at a rate of 0.00284% per year. If X is the time a randomly chosen plutonium atom will decay, write down the associated probability density function and use it to compute the probability that a plutonium atom will decay between 100 and 500 years from now.

Accepted Answer

f(x) = 0.0

Question 13

A certain solid S has a horizontal plane region R as its base and has height h cm measured perpendicular to R. For 0 3 cm3. Find (a) the area of the cross-section of S in the plane at height z cm and (b) the area of R.

Accepted Answer

A)  (a) 2 + 3z2 cm2, (b) 2 cm2 
B)  (a) 1 + 3z2 cm2, (b) 1 cm2 
C)  (a) 3 + z2 cm2, (b) 3 cm2 
D)  (a) 2 + 4z2 cm2, (b) 3 cm2 
E)  (a) 1 + 2z2 cm2, (b) 1 cm2 
A)  (a) 2 + 3z2 cm2, (b) 2 cm2 
B)  (a) 1 + 3z2 cm2, (b) 1 cm2 
C)  (a) 3 + z2 cm2, (b) 3 cm2 
D)  (a) 2 + 4z2 cm2, (b) 3 cm2 
E)  (a) 1 + 2z2 cm2, (b) 1 cm2

Question 14

Find the length of the closed loop part of the curve 3   = x   .

Accepted Answer

A)    units 
B)    units 
C)    units 
D)  2 units 
E)  1 unit 
A)    units 
B)    units 
C)    units 
D)  2 units 
E)  1 unit

Question 15

A vertical cylindrical container 10 m high and 6 m in diameter is half full of water. Find, to 4 significant figures, the amount of work done in pumping all the water out at the top of the container given that the density of water is 1,000 kg/m3.

Accepted Answer

A)  1.022 × 107 N . m 
B)  1.10 × 107 N . m 
C)  1.045 × 107 N . m 
D)  1.039 × 107 N . m 
E)  1.323 × 107 N . m 
A)  1.022 × 107 N . m 
B)  1.10 × 107 N . m 
C)  1.045 × 107 N . m 
D)  1.039 × 107 N . m 
E)  1.323 × 107 N . m

Question 16

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 y = -2 is V = $\pi$   dx.

Accepted Answer

A) True 
 B)False

Question 17

Use Pappus's Theorem to find the volume of the solid generated by revolving the region R enclosed by y =   , x = 0, and y =1 about the line y = -1 given that the centroid of the region R is at the point(   ,   ) = (   ,   ).

Accepted Answer

Volume = @#IMG-DLM&

Question 18

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

Accepted Answer

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

Question 19

the x-axis.

Accepted Answer

The answer of   the x-axis....

A random variable X is normally distributed with mean 100 and standard deviation 20. Below what value of X does 25% of the probability lie?

For what constant k if f(x) = k a probability density function on [0, 1]?

Find, correct to 4 decimal places, the length of the curve y = from x = 1 to x = 8.

Find the volumes of solids generated when the ellipse + = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

The base of a certain solid is a circular disk of radius a cm. Cross-sections of the solid in planes perpendicular to a specific diameter of the base are equilateral triangles. Find the volume of the solid.

Find the solution of the initial value problem = , y(π/2) = - .

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

Suppose the thickness of the blacktop on a road that has recently been resurfaced is normal with a mean of 6 cm and a standard deviation of 0.4 cm. What percentage of the blacktop is at least 5 cm thick?

Find the family of curves each of which intersects all of the hyperbolas x² - y² = C at right angles.

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.

Plutonium 239 decays continuously at a rate of 0.00284% per year. If X is the time a randomly chosen plutonium atom will decay, write down the associated probability density function and use it to compute the probability that a plutonium atom will decay between 100 and 500 years from now.

Find the length of the closed loop part of the curve 3 = x .

A vertical cylindrical container 10 m high and 6 m in diameter is half full of water. Find, to 4 significant figures, the amount of work done in pumping all the water out at the top of the container given that the density of water is 1,000 kg/m³.

Use Pappus's Theorem to find the volume of the solid generated by revolving the region R enclosed by y = , x = 0, and y =1 about the line y = -1 given that the centroid of the region R is at the point( , ) = ( , ).

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

the x-axis.

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

A random variable X is normally distributed with mean 100 and standard deviation 20. Below what value of X does 25% of the probability lie?

For what constant k if f(x) = k a probability density function on [0, 1]?

Find, correct to 4 decimal places, the length of the curve y = from x = 1 to x = 8.

Find the volumes of solids generated when the ellipse + = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

The base of a certain solid is a circular disk of radius a cm. Cross-sections of the solid in planes perpendicular to a specific diameter of the base are equilateral triangles. Find the volume of the solid.

Find the solution of the initial value problem = , y(π/2) = - .

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

Suppose the thickness of the blacktop on a road that has recently been resurfaced is normal with a mean of 6 cm and a standard deviation of 0.4 cm. What percentage of the blacktop is at least 5 cm thick?

Find the family of curves each of which intersects all of the hyperbolas x2 - y2 = C at right angles.

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.

Plutonium 239 decays continuously at a rate of 0.00284% per year. If X is the time a randomly chosen plutonium atom will decay, write down the associated probability density function and use it to compute the probability that a plutonium atom will decay between 100 and 500 years from now.

Find the length of the closed loop part of the curve 3 = x .

A vertical cylindrical container 10 m high and 6 m in diameter is half full of water. Find, to 4 significant figures, the amount of work done in pumping all the water out at the top of the container given that the density of water is 1,000 kg/m3.

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 y = -2 is V = π\piπ dx.

Use Pappus's Theorem to find the volume of the solid generated by revolving the region R enclosed by y = , x = 0, and y =1 about the line y = -1 given that the centroid of the region R is at the point( , ) = ( , ).

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

the x-axis.

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

Find the family of curves each of which intersects all of the hyperbolas x² - y² = C at right angles.

A vertical cylindrical container 10 m high and 6 m in diameter is half full of water. Find, to 4 significant figures, the amount of work done in pumping all the water out at the top of the container given that the density of water is 1,000 kg/m³.