Question 1

Where is the center of mass of a region of uniform density bounded by  ?

Accepted Answer

A)   
B)   
C)   
D)

Question 2

The pdf describing the chance that a given insect will die at a particular age, x days, is  . What is the chance the insect will live between 2 and 4 days?

Accepted Answer

A)  0.809 
B)  0.130 
C)  0.938 
D)  0.221

Question 3

A force of 6 pounds stretches a spring 3 inches. Find the work done in stretching this spring 4 inches beyond its natural length.

Accepted Answer

A)   foot-pounds 
B)   foot-pounds 
C)   foot-pounds 
D)   foot-pounds

Question 4

The base of a solid V is the region bounded by  and  Find the volume if V has semicircular cross sections perpendicular to the x-axis.

Accepted Answer

A)   
B)   
C)   
D)

Question 5

The thrust on a particular missile is represented as  Newtons for  second. Identify the correct impulse.

Accepted Answer

A)  0.0009 Ns 
B)  9,000 Ns 
C)  0.8 Ns 
D)  7.8 Ns

Question 6

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 40 feet. [Take the origin to be on the ground.]

Accepted Answer

A)  y(0) = 0, y'(0) = 0 
B)  y(0) = 40, y'(0) = 40t 
C)  y(0) = 0, y'(0) = -40 
D)  y(0) = 40, y'(0) = 0

Question 7

An electrical power supply, such as a battery, generates an electrical potential (voltage, V) which varies depending on how much current (i) it is delivering. The integral  represents the total power (energy per unit time) being given off by the battery when it is producing a current I. The useful electrical power, however, is just the product  . The difference between the total and the electrical power represents power that is wasted (and that accumulates as heat). Consider a hypothetical battery with a voltage vs. current curve described by the equation  . If the battery is producing a current of 4 amperes, how much electrical power is being produced, and how much power is being wasted?

Accepted Answer

A)   
B)   
C)   
D)

Question 8

Find the mean of the random variable with the given pdf.

Accepted Answer

A)   
B)   
C)   
D)

Question 9

Identify the mass of a thin wire with density  .

Accepted Answer

A)   
B)   
C)   
D)

Question 10

Find the volume of the solid formed by revolving the region bounded by  about y = 0.

Accepted Answer

A)   
B)   
C)   
D)

Question 11

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral  .

Accepted Answer

A)   
B)   
C)   
D)

Question 12

Find a value of c for which  is a pdf on the indicated interval.

Accepted Answer

A)   
B)   
C)   
D)

Question 13

Sketch and estimate the area determined by the intersections of the following curves using the method of trapezoids.

Accepted Answer

A)    
B)    
C)    
D)

Question 14

Molding is typically long, thin, decorative pieces of wood used, for instance, around windows or doors. The cross-section of a particular type of molding can be described as the region bounded by  , x = 4, x = 0, and  (all measured in inches), projected perpendicular to the xy-plane (see figure). If the manufacturer paints all four sides (but not the ends) and each gallon of paint covers 250 ft2, how many linear feet of molding can be coated with one gallon?

Accepted Answer

A)  315.8 feet 
B)  264.2 feet 
C)  512.4 feet 
D)  128.1 feet

Question 15

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places.  revolved about the x-axis

Accepted Answer

A)   
B)   
C)   
D)

Question 16

A race track is described by the equations  , and  . How long is one lap around the track? Round your answer to the nearest tenth.

Accepted Answer

A)  16.6 
B)  5.5 
C)  33.1 
D)  303.8

Question 17

A projectile traveling with momentum 2 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s2 and it is exerted over a time range  . What is the object's momentum (in the direction of the impulse) after the impulse?

Accepted Answer

A)  10 kg m/s 
B)  -22 kg m/s 
C)  -21 kg m/s 
D)  20 kg m/s

Question 18

Find the length of the line segment described by the function  between x = 1 and x = 4. Round your answer to three decimal places.

Accepted Answer

A)  3.550 
B)  3.350 
C)  3.15 
D)  2.950

Question 19

Find the median of x, given the pdf  . Round to the nearest integer.

Accepted Answer

A)  22 
B)  128 
C)  38 
D)  25

Question 20

A ball is thrown at an angle of  with an initial speed of 40 meters/second. What is its time of flight ? [Assume the ball is thrown from ground level and lands at ground level.]

Accepted Answer

A)  0.14 seconds 
B)  5.00 seconds 
C)  1.77 seconds 
D)  5.77 seconds

Exam 6: Applications of the Definite Integral

Where is the center of mass of a region of uniform density bounded by ?

The pdf describing the chance that a given insect will die at a particular age, x days, is . What is the chance the insect will live between 2 and 4 days?

A force of 6 pounds stretches a spring 3 inches. Find the work done in stretching this spring 4 inches beyond its natural length.

The base of a solid V is the region bounded by and Find the volume if V has semicircular cross sections perpendicular to the x-axis.

The thrust on a particular missile is represented as Newtons for second. Identify the correct impulse.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 40 feet. [Take the origin to be on the ground.]

Find the mean of the random variable with the given pdf.

Identify the mass of a thin wire with density .

Find the volume of the solid formed by revolving the region bounded by about y = 0.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Find a value of c for which is a pdf on the indicated interval.

Sketch and estimate the area determined by the intersections of the following curves using the method of trapezoids.

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places. revolved about the x-axis

A race track is described by the equations , and . How long is one lap around the track? Round your answer to the nearest tenth.

A projectile traveling with momentum 2 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s² and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

Find the length of the line segment described by the function between x = 1 and x = 4. Round your answer to three decimal places.

Find the median of x, given the pdf . Round to the nearest integer.

A ball is thrown at an angle of with an initial speed of 40 meters/second. What is its time of flight ? [Assume the ball is thrown from ground level and lands at ground level.]