Explanation Given: Surface function: f ( x , y ) = 25 − x 2 − y 2 Region below the surface: R = { ( x , y ) : x 2 + y 2 ≤ 25 } Formula used: Surface Area: S . A = S . A = ∬ R 1 + [ f x ( x , y ) ] 2 + [ f y ( x , y ) ] 2 Where, f x = ∂ f ( x , y ) ∂ x and f y = ∂ f ( x , y ) ∂ y x 2 + y 2 = r 2 Calculation: Consider the figure of region R = { ( x , y ) : x 2 + y 2 ≤ 25 } Consider the provided surface function: f ( x , y ) = 25 − x 2 − y 2 Partial derivative of above function with respect to x and y , f x ( x , y ) = − 2 x ; f y ( x , y ) = − 2 y Provided that x 2 + y 2 ≤ 25 This means r 2 ≤ 25 Or, 0 ≤ r ≤ 5 We don’t have to consider negative values of radius as radius is always non-negative. Insert the values into formula of surface area, S . A = ∬ R 1 + [ f x ( x , y ) ] 2 + [ f y ( x , y ) ] 2 = ∬ R 1 + ( − 2 x ) 2 + ( − 2 y ) 2 d x d y

11th Edition

ISBN: 9781337275576

Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

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Chapter 14, Problem 43RE

To determine

To calculate: Find the Surface Area specified by z=f(x,y) and located above the region R.

f(x,y)=25−x2−y2R={(x,y):x2+y2≤25}

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Chapter 14, Problem 42RE

Chapter 14, Problem 44RE

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Find the Surface Area specified by z = f ( x , y ) and located above the region R . f ( x , y ) = 25 − x 2 − y 2 R = { ( x , y ) : x 2 + y 2 ≤ 25 }

Solution Summary: The author calculates the Surface Area specified by z=f(x,y) and located above the region R.

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To calculate: Find the Surface Area specified by z=f(x,y) and located above the region R.

f(x,y)=25−x2−y2R={(x,y):x2+y2≤25}

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Chapter 14 Solutions

Find the Surface Area specified by z = f ( x , y ) and located above the region R . f ( x , y ) = 25 − x 2 − y 2 R = { ( x , y ) : x 2 + y 2 ≤ 25 }

Solution Summary: The author calculates the Surface Area specified by z=f(x,y) and located above the region R.

Concept explainers

Videos

To calculate: Find the Surface Area specified by z=f(x,y) and located above the region R. f(x,y)=25−x2−y2R={(x,y):x2+y2≤25}

Want to see the full answer?

Chapter 14 Solutions

To calculate: Find the Surface Area specified by z=f(x,y) and located above the region R.

f(x,y)=25−x2−y2R={(x,y):x2+y2≤25}