Explanation Given : f ( x , y ) = x y and the domain D in the figure Calculation : Step 1: The domain D can be described as a vertically simple region as follows: 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 − x 2 Step 2: The domain D can also be described as a horizontally simple region. To do this, we must express x in terms of y, for non negative values of x. This gives y = 1 − x 2 ⇒ x 2 = 1 − y ⇒ x = 1 − y Step 3: Hence we can describe D by the following inequalities: 0 ≤ y ≤ 1 , 0 ≤ x ≤ 1 − y Step 4: Computing the integral of f ( x , y ) = x y over D first using definition obtained in Step 1, we get ∫ ∫ D x y d A = ∫ 0 1 ∫ 0 1 − x 2 x y d y d x = ∫ 0 1 x y 2 2 | y = 0 1 − x 2 d x = ∫ 0 1 x 2 ( ( 1 − x 2 ) 2 − 0 2 ) d x = ∫ 0 1 x ( 1 − x 2 ) 2 2 d x = 1 2 ∫ 0 1

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ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 16.2, Problem 3E

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The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of $f (x, y) = x y$ over D as an iterated integral in two ways.

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Chapter 16.2, Problem 2E

Chapter 16.2, Problem 4E

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The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of f ( x , y ) = x y over D as an iterated integral in two ways.

Solution Summary: The author explains how the domain D can be described as a vertically and horizontally simple region. The value of the integral of f = x y over D is 112.

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The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of $f (x, y) = x y$ over D as an iterated integral in two ways.

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Chapter 16 Solutions

The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of f ( x , y ) = x y over D as an iterated integral in two ways.

Solution Summary: The author explains how the domain D can be described as a vertically and horizontally simple region. The value of the integral of f = x y over D is 112.

Videos

The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of f(x,y)=xyover D as an iterated integral in two ways.

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Chapter 16 Solutions

The description of the domain D in the figure as both, a vertically simple region and a horizontally simple region and the evaluation of the integral of $f (x, y) = x y$ over D as an iterated integral in two ways.