Question

1. (13 points) Use the limit of a Riemann Sum (i.e. sigma notation and the appropriate summation formulas) to evaluate the net-signed area between the graph of f(0) = 23 – 3 and the interval (0, 2). Let 27 be the right endpoint of the k-th subinterval (where all subintervals have equal width). Give your answer as a single integer or frac- tion, whichever is appropriate. Using any technique other than a limit of a Riemann Sum will earn no credit. Summation formulas: n n n Ek = n(n+1) 2 k2 k=1 n(n + 1)(2n +1) 6 n(n+1) 2 k=1 k=1 2. (14 points) Find the volume of the solid that results when the region enclosed by y = x² In x, y = 0, and x = e2 is revolved around the x-axis. Give a simplified exact answer, i.e. do not put in decimal form. 50+ 0

Answer 1

otch 2h 3h ith nių m Noso Sanof Prola of rectuples with right end point 1. Interval [0,2 YA 2-0=nh = n4=2 is divided in nequal part of each widthh 2-3 =nih fengan Xi > 1 2 =24 Moro hi nik x33 of our -3 efluith flulta-thfren) 거 shif lui) 2 2 top finiana (123) da = nuo 21 h 2 nyco lim & h (24²3) hlath-3) 3 the -3-6) =l nos le In nes 19 رها Nontº)-340 lim (ne) chhth Lunch 4127] u ntos h hoes ar ho =4-8 [412 4 (2th? X 4 Amy

2. y duanya da =lt nilinde Garisan » X Voleme=rehende dú ps . . sa هلا hin h In kan du = flux d (8) hnu-/ 20 dhun wolnu-s W lulu = - to sat od lla 25 s n حل ach no 25 26 w che - No lun-2 si dhe a 28 F askin 7107 3 asluinus 23 כן ") 25 e Volume nslun 2 é ohne 1 0). relo) ل 2 2 elo elo 25 lo loeloe elo ge = - 25 25 An.