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INSTRUCTIONS: This is a 3 hour exam. You are allowed the front and back of one 3 inch by 5 inch note card. Show all work for partial credit. Please ask if the statement of a question is unclear. Please put your name on EACH page of this exam.

Total accumulated points: out of 200 points.

1. Let R be the region bounded by the graphs of the functions f(x)=-2x(x-3) and g(x)=2x.
   1. (6pts) Set up the integral(s) that represent the area of R.
   2. (6pts) Set up the integral(s) that represent the volume of the solid generated by revolving the region R about the y axis.
2. (6pts) A force of 1000 pounds is required to compress a spring 5 inches from its natural length. Set up the integral for the work done in compressing the spring an additional 4 inches.
3. (7pts) A metal sheet in the shape of an isosceles triangle, with a base of 4 feet and a height of of 3 feet, is submerged vertically in water as indicated in the following diagram. Set up the integral that represents the fluid force on one side of th sheet. (recall that the weight-density of water is 64 pounds per cubic foot)

4. Find the following integrals.
   1. (6pts)
   2. (6pts)
   3. (6pts)
   4. (6pts)
   5. (6pts)
   6. (6pts)
   7. (6pts)
   8. (6pts)
5. Evaluate the following limits.
   1. (6pts)
   2. (6pts)
6. (6pts) Write an expression for the nth term of the sequence,

   

   and determine if the sequence converges.

7. Determine the convergence or divergence of the following series. If the series converges, then find the sum.
   1. (6pts)
   2. (6pts)
8. Determine the convergence or divergence of the following alternating series. If the series converges, then state whether or not it is absolutely convergent.
   1. (6pts)
   2. (6pts)
9. Determine the convergence or divergence of the following series.
   1. (6pts)
   2. (6pts)
   3. (6pts)
   4. (6pts)
10. (6pts) Determine both the radius of convergence and the interval of convergence for .
11. (6pts) Given the power series expansion for is

    

    find the Maclaurin series for

    

12. (7pts) Find the Taylor series expansion for

    

    centered at x=1.

13. Let .
    1. (6pts) Convert the above polar equation to rectangular form without inverse trig. functions. Do not worry about simplifying your answer.
    2. (6pts) Find the slope of the tangent line to the curve when .
    3. (6pts) Set up the integral representing the area enclosed by the graph.
14. Let x(t)=2t-2 and .
    1. (6pts) Find in terms of t.
    2. (6pts) For what value(s) of t does the graph have a horizontal tangent?
    3. (6pts) Set up the integral that represents the arc length of the curve from t=0 to t=2.
    4. (6pts) Eliminate the parameter t and express y in terms of x.