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Name: Math 2260 Final Exam May 2, 2012 Instructions: You are welcome to use one sheet of notes, but no other references or tools are allowed (no textbooks, no calculators, etc.). This is a 2 hour exam; you may start working at 8:00 am and must stop at 10:00 am. To receive full credit for a correct answer you must demonstrate how you arrived at that answer. To receive partial credit for an incorrect answer your work must be clearly explained. 1. (5 points) Consider the region bounded above by the graph of y = sin(x/2) and below by the x-axis. What is the volume of the solid obtained by revolving this region around the x-axis? 2 2. (5 points) Find the surface area of the cone given by rotating the line y = 2x for 0 ≤ x ≤ 4 around the x-axis. 3 3. (5 points) Solve the initial-value problem dy − xy = x, dx 4 y(0) = 3. 4. (5 points) Evaluate the integral Z e3 1 5 ln x √ dx. 3 x2 5. (5 points) Evaluate the integral Z 2 √ 0 6 1 dx. 4 − x2 6. (8 points) (a) Does the series ∞ X (−1)n n=1 n2 + π √ n3 − 2/2 converge absolutely, converge conditionally, or diverge? (b) Does the series ∞ X (n + 3)3 (n + 1)! n=1 converge absolutely, converge conditionally, or diverge? 7 7. (5 points) What is the interval of convergence of the following power series? ∞ X (−1)n (x − 2)n . n5n n=1 8 8. (5 points) Find an approximation for Z 0 1 sin(t2 ) dt t which is accurate to within 0.01. Feel free to give your answer as a fraction. [Hint: Think about Taylor series.] 9 * * * * * 9. (5 points) u = h1, 2i and v = h4, 2i are two vectors in the plane. Find vectors a and b so that a is * * * parallel to v, b is perpendicular to v, and * * * u = a + b. * u * * v b * a 10