2009 Exam3 Solution

2009 Exam3 Solution - Math 2263 Fall 2009 Midterm 3 WITH...

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Math 2263 Fall 2009 Midterm 3, WITH SOLUTIONS December 3, 2009 1. (10 points) Let B be the box, or rectangular solid: 0 x 4, 0 y 3, 0 z 1. Find ZZZ B (3 xz + 2 yz 3 - 5 x 2 z 4 ) dV. SOLUTION: ZZZ B (3 xz + 2 yz 3 - 5 x 2 z 4 ) dV = Z 1 0 Z 3 0 h 2 x 3 z + 2 xyz 3 - 5 3 x 3 z 4 i 4 x =0 dy dz = Z 1 0 Z 3 0 ± 16 z + 8 yz 3 - 5 3 · 64 z 4 ² dy dz = Z 1 0 h 16 y z + 4 y 2 z 3 - 5 3 · 64 yz 4 i 3 y =0 dz = Z 1 0 ± 48 z + 36 z 3 - 64 · 5 z 4 ² dz = h 32 z 3 / 2 + 9 z 4 - 64 z 5 i 1 z =0 = 32 + 9 - 64 = - 23 . 2. (15 points) Let S be the portion of the surface z = x 2 - y 2 inside the cylinder x 2 + y 2 = 9. Find A ( S ), the surface area of S . SOLUTION: The integrand of surface area is dS = q 1 + ( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 dA ( x,y ) , so let D be the disk x 2 + y 2 9 of radius 3: A ( S ) = ZZ D p 1 + 4 x 2 + 4 y 2 dA ( x,y ) = Z 2 π 0 Z 3 0 p 1 + 4 r 2 r dr dθ = 2 π Z 37 1 u 1 8 du = π 4 h u 3 / 2 3 / 2 i 37 u =1 = π 4 ± (37 37 - 1) ² .
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2009 Exam3 Solution - Math 2263 Fall 2009 Midterm 3 WITH...

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