Math 2242 Final Project Spring 2015 Due Date: 11:59 pm, May 5, 2015 Instructor: Jianqiang Zhao The Honor Code: On my honor, I will be academically honest in all of my course work and will not tolerate the academic dishonesty of others. I also pledge to engage in ethical behavior on-campus and o-campus, to live an honorable lifestyle, and to create a campus environment that is characterized by individual responsibility, civility, and integrity. Name: Signature: Name: Signature: Name: Signature: Directions: Collaboration is encouraged. You can work in groups. Each group can have at most three students. You need to show all the intermediate steps. NO CREDIT WILL BE GIVEN WITHOUT SUPPORTING WORK. You may get partial credits the problem. You should choose only ONE problem to work on. Submit only one set of answer in pdf format for each group via an email to jianqiangzhao@georgiasouthern.edu. You can rst write your answer by hand, scan it or take a picture, then convert it to pdf. 1. One way to nd the sum of an innite series is to regard it as a special value of a power series. For example, 1 1 = nxn1 . = f (x) = 2 (1 x) 1x n=1 Thus taking x = 1/2 we get n=1 n 2n1 = f (1/2) = 4. For more examples, please study 10.10 (and its exercises) of our textbook. Use this idea to nd all of the following innite sums: 2n+1 (1)n (1)n 3 (2n 1)!! 1 (a). , (b). , (c) 2n+1 2n n 2n + 1 (2n)!! 2 (2n + 1) n=1 n=0 n=1 where k!! = k (k 2)!! for every positive integer k 2, 0!! = 1!! = 1. 2. For this project, you need to study 11.2 (and its exercises) of our textbook. Then solve all of the following three parts. x2 y 2 (a) Find the area enclosed by the ellipse + = 1. 2 18 (b) Find the coordinates of the centroid of the curve x = ex cos t, y = ex sin t, t [0, ]. (c) Find the area of the surface generated by revolving the curve x = (2/3)t3/2 , y = 2 t, t [0, 3] around y-axis