1. Use Pappus' Theorem to calculate the volume generated when rotating the region bounded by y = 2 -x, y = 0, and r = 0 about the r-axis. No points will be awarded for another method. 2. A certain basketball has the property that when it falls from height H it rebounds to height . H. If the ball is initially dropped from a height of 2 meters find the total distance it travels, assuming it continues to bounce indefinitely. 3. Evaluate the following integral: -3r' +1-1 23 + 2x2 + x +2 dx [Hint: denominator is zero if r = -2.] 4. Let F(s) = / f(tje at dt. Find F(s) if f(t) = t. 5. [20 points] Consider the polar curve r = 2 + cos(0). (a) Analytically (non-graphically) find all points on the graph of the curve with vertical tangents. (b) Sketch the graph by first creating a Cartesian graph then graphing the polar curve. 6. Find the radius and interval of convergence of the power series: - (x - 2)" 10 7. Whispering galleries are rooms designed with (semi-) elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center. 8. [15 points] Assume the power series ) and, converges to a function f(x) such that f(0) = 1, f(0) = 0, and f" (x) = f(x). Find f(x) by the following procedure: [a) Equate the corresponding terms of the power series of f"(x) with those of f(x). Write down the recursive relationship between on and an-2. (b) Use the facts that /(0) = 1, f'(0) = 0, and your relation from (a) to make a table of an for n = 1,2, . . . ,9. [Hint: do not evaluate the products of numbers.] (c) Write out /(x) as a power series. Include at least 5 non-zero terms and a summation formula. 9. Evaluate the following integral: COG (I) V1 + sin (x)