100hw5soln

# 100hw5soln - Math 100 Homework 5 fall 2010 1 Let f be a...

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Unformatted text preview: Math 100 Homework 5 fall 2010 1. Let f be a function in two variables defined by f ( x, y ) = 1- x , R is the region consisting of points ( x, y ) satisfying y 2 ≤ x ≤ 1 and- 1 ≤ y ≤ 1. Then, f ( x, y ) ≥ 0 when ( x, y ) belongs to R , and the given solid is the collection of points ( x, y, z ) so that ( x, y ) belongs to R and 0 ≤ z ≤ f ( x, y ). Its volume is R R f = R 1- 1 R 1 y 2 (1- x ) dxdy = R 1- 1 [1- y 2- 1 2 (1- y 4 )] dy = 8 / 15 . 2. Let R be the region collecting points ( x, y ) satisfying 0 ≤ y ≤ ln x and 1 ≤ x ≤ e . Then, the given integral is infact R R f . Alternatively, this region R collects points ( x, y ) satisfying e y ≤ x ≤ 1 and 0 ≤ y ≤ 1. Therefore, Z e 1 Z ln x f ( x, y ) dydx = Z R f ( x, y ) dxdy = Z 1 Z 1 e y f ( x, y ) dydx. 3. Let R be the region collecting points ( x, y ) satisfying- p 4- y 2 ≤ x ≤ p 4- y 2 and- 2 ≤ y ≤ 2. That is, R is the disc with radius 2 centered at the origin. It collects points whose polar coordinates (at the origin....
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## This note was uploaded on 12/25/2011 for the course MATH 101 taught by Professor Ching during the Spring '11 term at HKU.

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100hw5soln - Math 100 Homework 5 fall 2010 1 Let f be a...

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