MAT 1575 Final Exam Review Problems Revised by Prof. Kostadinov Spring 2014, Prof. ElHitti Summer 2017, Prof. Africk Spring 2023 1. Evaluate the following definite integrals: a. fox2(x3 + 1)3 dx b . X dx C. 3x2 = dx Vx2 + 9 oux' + 1 2. Evaluate the following indefinite integrals: a. [x2 In(x) dx b. [xe * dx c. [xcos(3x) dx 3. Find the area of the region enclosed by the graphs of: a. y= 3-x and y=-2x b. y=x -2x and y = x +4 4. Find the volume of the solid obtained by rotating the region bounded by the graphs of: a. y= x' -9, y =0 about the x-axis. b. y = 16-x, y= 3x+12, x=-1 about the x-axis. c. y=x2 + 2, y= -x2 + 10, x 2 0 about the y-axis. 5. Evaluate the following indefinite integrals: 9 6 a. x2 36- x2 dx b . [ x 2 - 9 dx c . dx d. dx x2 Vx2 + 9 x2 Vx2 - 36 6. Evaluate the following indefinite integrals: 3x+7 a. J 3x+ 2 x2+6x+9 dx ( 5x + 6 x 6. 2 - 36 C. J 72 +2 x-8 -dx 7. Evaluate the improper integral: 2 dx b . 5 3 a. Ex + 5 dx (x + 2 ) 3 3 ( x -3) 4 dx 8. Decide if the following series converges or not. Justify your answer using an appropriate test: Sn=oo ins b . 5 d. En=co n! n+1 a. Zn=1 35+5 e. En=oo 210" c. Sn 10" Zn=1 n250 =1 2n+3. 9. Determine whether the series is absolutely or conditionally convergent or divergent: E(-1)"- 10 b . c. [(-1) "5-" d. [ (- 1)" n - n- 1 71=1 7n + 2 1 0 2n +n+ 1 10. Find the radius and the interval of convergence of the following power series: a. (x - 1)n b . (-1)"(x - 1)n (x +1)n M C . d. (-1)" (x + 1)n n=0 n+ 2 n5n n=0 n + 2 n5n n=1 n=1 1 1. Find the Taylor polynomial of degree 2 for the given function, centered at the given number a: a. f(x) = e * at a=-1. b. f(x) = cos(5x) at a = 2x. 12. Find the Taylor polynomial of degree 3 for the given function, centered at the given number a: a. f(x) = 1+ e * at a=-1 b. f(x) = sin(x) at a =~