Summer Semester 2016 MATH 22: Calculus II - Final You have 1 hr. 45 min. to complete this exam. Show all of your work, simplify your answers fully and box them. There are no notes, no calculators, no electronic devices whatsoever permitted on this exam. 1. (20 pts) Find the volume obtained when the area bounded by y = 6, y = 5 + 2 cos x, and the y-axis for x > 0 is rotated about the y = 5 axis. 2. (20 pts) Find the points on the polar curve r = 10 sin where the tangent vector is vertical or horizontal. Express your answers as points in rectangular coordinates. 3. (80 pts - 20 each) Compute the integral. Z 1 (a) dx x x2 9 Z 1 (b) dx x2 2x 15 Z (c) 8 Z (d) 1 dx 3 x x2 e x3 ln xdx 1 4. (40 pts - 20 each) Find the interval and radius of convergence of the power series. X (x 3)n (n + 2)10n n=1 (a) (b) X 7n (5x + 4)n n! n=1 5. (40 pts - 20 each) Find a Maclaurin series for each of the following functions: Z x 2 (a) f (x) = x3 ex (b) f (x) = cos t2 dt 0 Formulas You May Find Useful b Z V = r22 r12 \u0001 dx A= b r 1+ s= \u0010 dy \u00112 dx dx \u0010 dr \u00112 s= r2 + d d Z t2 r \u0010 \u0011 2 \u0010 \u0011 2 dx dy s= + dt dt dt t1 A= a Z y(t) a a Z b Z r 1 2 Z dx dt dt |Rn (x)| bn+1 ex = 2 (r()) d X xn n! n=0 Z tan xdx = ln |sec x| + C sin x = X (1)n x2n+1 (2n + 1)! n=0 cos x = X (1)n x2n (2n)! n=0 Z sec xdx = ln |sec x + tan x| + C 1