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Unformatted text preview: Math136 Review for Final Exam Summer 2010 1. For each one of the regions R described below, write and evaluate an integral expressing: (a) the area or R . (b) the volume of the solid resulting when revolving R about the x- axis. (c) the volume of the solid resulting when revolving R about the y- axis. i. Enclosed by the curves y = ( x- 2) 2 and x + y = 4 . ii. cos x y sec 2 x, x / 4 iii. Below y = x , above the line y- x + 2 = 0 and the x- axis. 2. Compute the improper integrals (a) Z e 3 x ( e 3 x + 15) 5 / 4 dx (b) Z / 2 sin x ln(cos x ) dx 3. (a) e 2 x dy dx + x cos 2 y = 0 , y (0) = 0 (b) dy dx + 2 xy 1 + x 2 = sin x, y (0) = 1 4. The base of a solid S is the region on the xy plane enclosed by the ellipse x 2 + 3 y 2 = 1. Its cross sections perpendicular to the y- axis are isorectangle triangles (base=height). Find the volume of the solid S . 5. Let I = R 4 1 x 5 / 2 dx . (a) Use the trapezoidal rule with n = 6 to give an approximation to I ....
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- Spring '08