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Problem Set 9 Solution

Problem Set 9 Solution - MODEL ANSWERS TO HWK#9 1 There are...

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�� MODEL ANSWERS TO HWK #9 1. There are a number of ways to proceed; probably the most straight- forward is to view the region D as something of type 2: �� 2 �� 2 y x + y d x d y = x + y d x d y 2 2 y D 1 2 y 2 y 2 x = + yx d y 1 2 2 2 y y 2 2 2 = 1 (2 2 y ) 2 + y (2 y ) ( y 2 2 y ) 2 y ( y 2 y ) d y 2 4 2 = y 2 + y 3 y 2 + 2 d y 1 5 4 3 2 y y y = + + 2 y 2 5 4 2 3 · · 1 2 4 2 2 1 1 1 = 5 + 2 2 3 + 2 2 10 4 6 + 2 99 = . 20 2. There are a number of ways to proceed; probably the most straight- forward is to view the region D as something of type 1: 1 9 �� 3 x 1 / 2 1 3 y d x d y = 3 y d y d x + 3 y d y d x D 0 x 1 9 x 2 3 1 2 x 1 / 2 1 9 3 y 3 y d x + d x = 2 2 x 1 9 0 x 1 3 3 3 x 2 3 3 x 2 d x + d x 1 9 2 2 x = 2 2 1 9 0 3 1 1 9 3 3 x x 3 3 x = 2 2 0 + 2 ln x 2 1 9 3 1 1 1 = + 3 ln 3 + 2 2 3 6 2 2 3 6 · · = 1 + 3 ln 3 . 1
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3. 2 4 y 2 2 4 y 2 2 x x d x d y = d y 2 0 0 0 0 2 (4 y 2 ) 2 = d y 2 0 2 4 = 8 4 y 2 + y d y 2 0 4 y 3 y 5 2 = 8 y 3 + 2 5 · 0 32 2 4 = 16 + 3 5 2 4 3 5 2 5 5 + 2 4 3 = · · · · 3 5 · = 2 4 · 3 · 6 2 5 · 5 3 5 · 2 5 (9 5) = 3 5 · 2 7 = . 3 5 · The region in question is bounded by the curves x = 0, y = 0 and y 2 = 4 x . So, reversing the order of integration, we get 4 4 x 4 4 x x d y d x = x [ y ] 0 d x 0 0 0 4 = x 4 x d x 0 4 4 2 x 2 = 3
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