M2_2009_Math124 - MATH124 CALCULUS II for Engineers Midterm...

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MATH124 CALCULUS II for Engineers Course Section: Midterm 2 (50 pts) March 25, 2009 Please print names and IDs in ink : NSID: Family Name: First Name: Student ID: INSTRUCTIONS: 1 . Time Limit: 80 minutes 4 . Closed book. Closed notes. 2 . No cheating. No calculators. 5 . Write clearly & legibly. 3 . Simplify all answers unless otherwise instructed. SECTION I 30 pts Solve each of the following questions. 1. (6 pts) Evaluate the given integrals, or show that they diverge: (a) (3 pts) 2 1 ( x 1) 3 dx Solution : 2 1 ( x 1) 3 dx = lim b →∞ b 2 1 ( x 1) 3 dx = lim b →∞ 1 2( x 1) 2 b 2 = = lim b →∞ 1 2( b 1) 2 1 2 = 0 1 2 = 1 2 (b) (3 pts) 0 1 1 x + 1 dx = Solution : 0 1 1 x + 1 dx = lim a →− 1 + 0 a 1 x + 1 dx = lim a →− 1 + (ln | x + 1 | ) 0 a = = lim a →− 1 + (ln 1 ln | a + 1 | ) = 0 ( −∞ ) = + 2. (4 pts) Find the volume of the solid obtained by rotating the region bounded by y = cos x , x = 0, y = 0, x = π 2 , about the x -axis. Solution : V = π / 2 0 π ( cos x ) 2 dx = π π / 2 0 cos x dx = π sin x π / 2 0 = = π (sin( π / 2) sin 0) = π (1 0) = π 3. (4 pts) Find the volume of the solid obtained by rotating the region bounded by y = 2 x x 2 and y = 0 about the y -axis. Solution : V = 2 0 2 π x (2 x x 2 ) dx = 2 π 2 0 (2 x 2 x 3 ) dx = 2 π 2 x 3 3 x 4 4 2 0 = = 2 π 2 · 8 3 16 4 = 2 π 16 3 16 4 = 2 π · 16 12 = 8 π 3
4. (4 pts) Find the length of the curve