# a10 solu - MATH 251-3 Fall 2007 Simon Fraser University...

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Unformatted text preview: MATH 251-3, Fall 2007 Simon Fraser University Assignment 10: Solutions Additional Question Due: 4:30pm, Wednesday 5 December 2007 1. Let E be the part of the sphere of radius a in the first octant; that is, x 2 + y 2 + z 2 = a 2 , x ≥ 0, y ≥ 0, z ≥ 0. (a) The volume of E is given by the triple integral RRR E dV . Find the appropriate limits and write down this triple integral in (i) rectangular, (ii) cylindrical and (iii) spherical coordinates. Evaluate the triple integral in cylindrical and spherical coordinates. Does your answer agree with your expectations based on the formula for the volume of a sphere? Solution: (i) In rectangular coordinates, the region is x 2 + y 2 + z 2 = a 2 , whose projection onto the xy plane ( z = 0 ) is x 2 + y 2 = a 2 ; also, x ≥ , y ≥ , z ≥ . Integrating first with respect to z , then y , then x , the volume is given by the triple integral (rectangular) Z a Z √ a 2- x 2 Z √ a 2- x 2- y 2 dz dy dx ; this integral is rather cumbersome to evaluate. (ii) In cylindrical coordinates, with r 2 = x 2 + y 2 the formula for the sphere is r 2 + z 2 = a 2 , so ≤ z ≤ √ a 2- r 2 . The projection of the sphere is the quarter circle of radius a , in the first quadrant, so ≤ r ≤ a and ≤ θ ≤ π/ 2 . Remembering that the element of volume in cylindrical coordinates is dV = r dz dr dθ , the volume of E is given by the triple integral (cylindrical) Z π/ 2 Z a Z √ a 2- r 2 r dz dr dθ . Evaluating the integral, we get Z π/ 2 Z a Z √ a 2- r 2 r dz dr dθ = Z π/ 2 Z a r √ a 2- r 2 dr dθ = Z π/ 2 dθ- 1 2 · 2 3 ( a 2- r 2 ) 3 / 2 a = π 2 1 3 a...
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a10 solu - MATH 251-3 Fall 2007 Simon Fraser University...

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