prac-exam2-sol

prac-exam2-sol - Math 21B UC Davis, Winter 2010 Dan Romik...

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Math 21B UC Davis, Winter 2010 Dan Romik Solutions to practice questions for midterm exam 2 1. Let C denote the curve y = 2 - | x | , - 1 x 1. (a) Compute the volume of the solid of revolution formed by revolving the region bounded between C and the x -axis about the x -axis. (b) Compute the surface area of the surface of revolution of C about the x -axis. (c) Compute the arc length of C . Solution. In all three cases, we can use symmetry by computing the answer for the part of the curve in the range 0 x 1 and multiplying it by 2. This gives (a) V = Z 1 - 1 πy 2 dx = 2 Z 1 0 πy 2 dx = 2 π Z 1 0 (2 - x ) 2 dx = 2 π - (2 - x ) 3 3 ± ± ± 1 0 = 2 π 3 (2 3 - 1 3 ) = 14 π 3 , (c) L = 2 Z 1 0 s 1 + ² dy dx ³ 2 dx = 2 Z 1 0 2 dx = 2 2 , (b) A = 2 Z 1 0 2 π (2 - x ) 2 dx = ... = 6 2 π. 2. The parametric curve x = 2 3 t 3 / 2 , y = 2 t, 0 t 3 is revolved about the y -axis. Find the area of the resulting surface. 1
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Solution. The differential arc-length is given by ds = p dx 2 + dy 2 = s ± dx dt ² 2 + ± dy dt ² 2 dt = t + t - 1 dt = r t 2 + 1 t dt, so the surface area is A = Z 3 0 2 πx ( t ) r t 2 + 1 t dt = Z 3 0 2 π · 2 3 · t t 2 + 1 dt This integral can be computed by making the substitution u = t 2 + 1, leading to A = 2 π 3 Z 4 1 udu = ... = 28 π 9 . 3.
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prac-exam2-sol - Math 21B UC Davis, Winter 2010 Dan Romik...

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