NAME                                      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  . A)   (10 Points)   Find the derivative of sec-1(2x + 1) Solution B)    (10 Points) Show that           Solution   PROBLEM 3 Find the following indefinite integrals A)   (15 Points)       Solution B)    (15 Points)       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