MATH 201 Final, Spring 2004

American University of Beirut Calculus and Analytic Geometry Spring 2004 Final Exam Date: Monday, May 31, 2004 - 3:00 pm to 5:00 pm Instructors: Ms Sylvana Jaber, Mr Zadour Khachadourian and Dr. Mohamed Kobeissi 4 (Ms Jaber) 5 (Mr Khachadourian) 6 (Mr Khachadourian) T 3:30-4:20 pm T 2:00-2:50 pm T 12:30-1:20 pm Section: This is NOT an open-book exam. Your exam should have 11 pages including this one, and there are 8 questions totaling 100 points. You can continue each exercise on the reverse side if needed. "1 Question | Grade ; 1 anne emma neers eet fag cin AN ENINERSET rod LIBRARY | gh TET { Good luck n Exercise 1 {72 points}. a. yo (1- 5) +00 ol b. > msin — n n=l +00 c. S e "cosn n=1 4 = OO l r d. y = —< /n In? n - Determine if the following series converge or diverge. Justify your answers is lune ommend MERU yyy SIMEBSET LikRARY Nanay arene nnd Exercise 3 /16 points}: Let V be the volume of the smaller region cut from the bottom of the cone z= \/xr? — y* by the plane z = 3. a. Express V as an iterated triple integral in spherical coordinates (that is, set up the limits of integration but do not evaluate the resulting integral). b. Express V as an iterated triple integral in cylindrical coordinates, then evaluate the resulting integral to find V. c. Find the plane z = c that divides the region into two parts of equal volume. Exercise 4 //4 points}: Find the absolute minimum and maximum for the function f(z,y) = a2? + 4y? — y on the region R = {(x,y): 2? + 4y? < ly > 0}. Exercise 5 [10 points]: Let V be the volume of the region whose vertices are (0, 0,0}, (1, 2.0), (1,2, 1), (0, 2,0) and (0, 2,1). Express V as an iterated triple integrals in the foliowing orders a. order dzdxdy (do not evaluate the integral) b. order dydzdx, then evaluate the integral Exercise 6 [12 points]: Evaluate the integral “Lo ple / [ * e-E4¥) dndy 0 vy by applying the transformationu = a-—yandu=z+y. wo phboy. Exercise 7 {16 points}: a, Find the value of the line mtegral (2.7 /2,1) / (32? + Qaz*)dx + (7zcosy)dy + (2272 + 7siny + e*)dz a 10,0) b. Find the counterclockwise circulation of the field F (x.y) = (a + e” siny)i+ (x + e* cos y)j around the triangle whose vertices are (0,0), (1,0) and (0,1)