<p> MATHS 102 Final Exam</p><p>University of Bahrain College of Science Department of Mathematics Summer Semester 2004/2005 Final Examination</p><p>MATHS 102 : Calculus II Date : Tuesday, 23rd August 2005 Time : 11:30-13:30 Max. Marks : 50 Examiners : Dr.A. AbuSafiya (Sec.1), Dr. S. Abbas (Sec.2), Dr. F. Al-Showaikh* (Sec.3) and Dr. A. Al-Mannai (Sec.4) </p><p>Name: I.D.No: Section:</p><p>Instructions to Candidates:</p><p>1- Write your name, I.D number and section number in the above box. 2- Make sure that you have 9 different pages and 5 questions. 3- If more space is needed, use the back of the previous page.</p><p>Marking Scheme</p><p>Question Max.Marks Marks Obtained 1 12 2 10</p><p>3 8 4 10 5 10 Total 50</p><p>* Coordinator</p><p>Good Luck</p><p>1 MATHS 102 Final Exam</p><p>Question 1: [ 12 Marks] a) Set up integrals that can be used to find the volume of the solid generated by revolving the region bounded by the graphs of y  x2 and y  2x  3  0 about: [6] i) the line x  3 ; ii) the line y  9 .</p><p>(DO NOT EVALUATE THE INTEGRALS)</p><p>2 MATHS 102 Final Exam b) Find the area of the region that is inside the circle r  1 and outside the cardiod r  1 cos . [6]</p><p>3 MATHS 102 Final Exam</p><p>Question 2: [ 10 Marks] </p><p>Evaluate the following integrals: a)  xsin 1 x dx [5]</p><p>1 b) dx [5]  2  3x  2x2</p><p>4 MATHS 102 Final Exam</p><p>Question 3: [ 8 Marks] a) Determine whether the following improper integral converges or diverges:</p><p>0 (xe x )  e dx [4] </p><p> tanh x b) Evaluate lim (sinh x) [4] x0</p><p>5 MATHS 102 Final Exam</p><p>Question 4: [ 10 Marks]</p><p>Determine whether the following series is absolutely convergent (A.C.), conditionally convergent (C.C.), or divergent:</p><p> k 1 1 a)  (1) k 1 k [5]</p><p>6 MATHS 102 Final Exam</p><p> k k! b) (1) [5]  k k 1 5</p><p>7 MATHS 102 Final Exam</p><p>Question 5: [ 10 Marks] a) Use a known Taylor polynomial with three nonzero terms to estimate the value of the 1 x3 integral  xe dx . [5] 0</p><p>8 MATHS 102 Final Exam</p><p> k k 1 (3x 1) b) Find the radius and interval of convergence of the power series (1) [5]  2 . k 1 k</p><p>9</p>