53 Worksheet 10_19 Solutions

# 53 Worksheet 10_19 Solutions - Math 53: Multivariable...

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Math 53: Multivariable Calculus Worksheet 10/19/11: Finding Volumes In Antarctica (Polar Integration) Exercise 0.1. Find the volume bounded by the paraboloids z = 3 x 2 + 3 y 2 and z = 4 - x 2 - y 2 . Solution. These paraboloids intersect at 3 x 2 +3 y 2 = 4 - x 2 - y 2 , which is just x 2 + y 2 = 1; call the region bounded by this circle in the xy -plane D . The volume between them is the volume under z = 4 - x 2 - y 2 and above D , minus the volume under z = 3 x 2 + 3 y 2 and above D . The ﬁrst volume: ZZ D (4 - x 2 - y 2 ) dA = Z 2 π 0 Z 1 0 (4 - r 2 ) rdrdθ = Z 2 π 0 Z 1 0 (4 r - r 3 ) drdθ = Z 2 π 0 ± 2 r 2 - r 4 4 ²³ ³ ³ ³ 1 0 = Z 2 π 0 7 4 = 7 4 θ ³ ³ ³ ³ 2 π 0 = 7 π 2 The second volume: ZZ D (3 x 2 + 3 y 2 ) dA = Z 2 π 0 Z 1 0 3 r 2 rdrdθ = Z 2 π 0 3 r 4 4 ³ ³ ³ ³ 1 0 = Z 2 π 0 3 4 = 3 4 θ ³ ³ ³ ³ 2 π 0 = 3 π 2 . The volume we’re interested is the ﬁrst minus the second, i.e. 7 π 2 - 3 π 2 = 2 π. ± 1

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2 Exercise 0.2. Convert to polar coordinates and then evaluate the integral Z a 0 Z 0 - a 2 - y 2 x 2 ydxdy Solution. Going through the bounds on
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## 53 Worksheet 10_19 Solutions - Math 53: Multivariable...

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