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MATH 106 PRACTICE MIDTERM 2

 

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

  Printable Key

PROBLEM 1  Set up the integrals that solve the following problems.  Sketch the appropriate diagram for each.  Then use a calculator to finish the problem.

A)   (10 Points) Find the volume of the solid that is formed by revolving the region bounded by y  =  x3 + x and y  =  x2 + x  around the y-axis.
Solution

B)    (10 Points). Find the volume of the solid that is formed by revolving the region bounded by  y  =  x4  and y  =  x between x  =  0 and  x  =  1/2  about the line y  =  -10.
Solution

C)   (10 Points) Find the volume of the solid that is formed by revolving the region bounded by y  =  x3 - x  and y  =  3x around the line x  =  5.
Solution

D)   (10 Points) Find the area of the region bounded by the curves
        y  =  x3 - 3x2 - 9x + 12 and y  =  x + 12
Solution

E)    (10 Points) Find the length of the curve y  =  sin x for  0 <   x  < 2p.
    Solution

F)    (10 Points) Find the volume of the sphere of radius 2.
    Solution

 

PROBLEM 2  You have been called as an expert witness in the case involving the recent murder that occurred in room E106.  It is clear that at the time of death the victim was healthy with a temperature of 98.6 degrees.  It is also know that a human body in this situation will cool down to 90 degrees in one hour.  When the body was discovered at 10:00 PM the corpse had a body temperature of 85 degrees.  During the entire day, the temperature of the room was a constant 65 degrees. 

A) (10 points) Use the Newton’s Law of Cooling (the rate of change of the temperature of the body is proportional to the difference between the body’s temperature and the ambient temperature) to write down a differential equation for this situation.  

Solution

B) (10 points) Solve the differential equation from part A.

Solution

C) (10 points) What was the time of death?

Solution

 

PROBLEM 3

Solve the following differential equations

A.  (15 Points) y(1 + x2)y'  -  x(1 + y2)  =  0,    y(0)  = 

Solution

B.  (15 Points)                 x3 + y3
                          y'  =                  
                                           xy2  

Solution

 

Problem 4   (20 Points)

When completed, the International Space Station orbiting at 238 miles above the surface of the earth will weigh one million pounds (at the surface of the earth 4000 miles from the center).  How much total work will it take to send the entire station in orbit?
    Solution

 

Problem 5  (20 Points)

A 30 foot chain that weighs 3 pounds per foot is used to lift a 200 pound piece of sheet metal from the ground to the top of a 30 foot tall building.  How much work is required?
    Solution