Explanation Given: The function f ( x , y ) = 16 − x 2 − y 2 Formula used: Average value over a plane R is given by A v g = 1 A ∬ R f ( x , y ) d A Here, A is the area of region R. Use the integration formula ∫ x n d x = x n + 1 n + 1 Calculation: Bounds for x is given as − 2 ≤ x ≤ 2 Bounds for y is given as − 2 ≤ y ≤ 2 Plot the graph of region R from given coordinates ( 2 , 2 ) , ( − 2 , 2 ) , ( − 2 , − 2 ) and ( 2 , − 2 ) The figure of above rectangle is shown below: Area of the obtained rectangle is: ( 2 − ( − 2 ) ) ⋅ ( 2 − ( − 2 ) ) = 16 unit 2 Put the value of function, limits and area in above formula, 1 16 ∫ − 2 2 ∫ − 2 2 ( 16 − x 2 − y 2 ) d y d x = 1 16 ∫ − 2 2 [ 16 y − x 2 y − y 3 3 ] − 2

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14, Problem 21RE

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To calculate: The average value of f(x,y) over a plane region R.

f(x,y)=16−x2−y2

R: rectangle with vertices (2,2), (−2,2), (−2,−2) and (2,−2).

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Chapter 14, Problem 20RE

Chapter 14, Problem 22RE

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The average value of f ( x , y ) over a plane region R. f ( x , y ) = 16 − x 2 − y 2 R : rectangle with vertices ( 2 , 2 ) , ( − 2 , 2 ) , ( − 2 , − 2 ) and ( 2 , − 2 ) .

Solution Summary: The author calculates the average value of f(x,y) over a plane region R.

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To calculate: The average value of f(x,y) over a plane region R.

f(x,y)=16−x2−y2

R: rectangle with vertices (2,2), (−2,2), (−2,−2) and (2,−2).

Want to see the full answer?

Chapter 14 Solutions

The average value of f ( x , y ) over a plane region R. f ( x , y ) = 16 − x 2 − y 2 R : rectangle with vertices ( 2 , 2 ) , ( − 2 , 2 ) , ( − 2 , − 2 ) and ( 2 , − 2 ) .

Solution Summary: The author calculates the average value of f(x,y) over a plane region R.

Videos

To calculate: The average value of f(x,y) over a plane region R. f(x,y)=16−x2−y2 R: rectangle with vertices (2,2), (−2,2), (−2,−2) and (2,−2).

Want to see the full answer?

Chapter 14 Solutions

To calculate: The average value of f(x,y) over a plane region R.

f(x,y)=16−x2−y2

R: rectangle with vertices (2,2), (−2,2), (−2,−2) and (2,−2).