Calculus II Final Exam Integrate the following by integration by parts

Calculus II Final Exam Integrate the following by integration by parts. 1 x 1) ∫ xe 2 dx All problems 10 points unless indicated Evaluate the integral using trig substitutions 2) ∫ x 2dx 9 − x2 Evaluate the integral using partial fractions 3) ∫ 2x + 1 dx x − 7 x + 12 2 4) Use the washer/disk method to find the volume of the solid generated by revolving the region bounded by the following curves about the x – axis: y = 4 – x2, and y = 2 – x 5) Use the shell method to find the volume of the solid generated by revolving the region bounded by the following curves about the y - axis: y = 12 – x, and y = x2 and to the right of x = 0. 6) Find the arc length of y = x3 1 + , 1 ≤ x ≤ 3 . Set up integral and use calculator to find the value. 6 2x Round answer to 4 decimal places y x + 1 , from 1 < x < 5, . 7) Find the area of the surface generated by rotating the curve, = about the x – axis. Set up integral and use calculator to find the value. Round to 4 decimal places Determine if the following series converges or diverges using the Integral Test 8) ∞ 1 ∑ n(ln n) n =2 15 points 9) For the following series ( x − 1) n , 3 n n =1 n 3 ∞ ∑ 15 pts a) Find the series’ radius and interval of convergence b) Check endpoints for any values that might be conditionally convergent 10) Find the first 5 terms and then the nth term of the Taylor series generated by f(x) = 1/x at a = 2. Then find a series representation 15 points 11) For the following parametric equations: = x 1 1 π = tan(t ) y = sec(t ) t 2 2 3 Find: a) The slope of the tangent line at any point. b) The slope of the tangent line at t. c) The equation of the tangent line at t. 15 points 12) For the following polar curve r = sin(3θ), find the exact answer for: a) the slope of the tangent line θ = π/2. b) the slope of the tangent line at θ = π/6. c) the total area enclosed by the curve. (Hint one leaf goes from θ = 0 to θ = π