Explanation Given: The surface is given by f ( x , y ) = a 2 − x 2 − y 2 , above the region R , represented by the limit shown R = { ( x , y ) : x 2 + y 2 ≤ a 2 } Formula used: The surface area can be calculated of the region R by, S = ∬ R 1 + [ f x ( x , y ) ] 2 + [ f y ( x , y ) ] 2 d A Differentiation formula d d x ( x n ) = n x n − 1 , d d x ( constant ) = 0 The equation of circle x 2 + y 2 = r 2 , where r is the radius Calculation: The function given is f ( x , y ) = a 2 − x 2 − y 2 . Now partially differentiating it with respect to x , use d d x ( x n ) = n x n − 1 , d d x ( constant ) = 0 f x ( x , y ) = d d x ( a 2 − x 2 − y 2 ) = − 2 x 2 a 2 − x 2 − y 2 = − x a 2 − x 2 − y 2 Now, with respect to y It is found that: f y ( x , y ) = d d x ( a 2 − x 2 − y 2 ) = − 2 y 2 a 2 − x 2 − y 2 = − y a 2 − x 2 − y 2 Putting in formula S = ∬ R 1 + [ − x a 2 − x 2 − y 2 ] 2 + [ − y a 2 − x 2 − y 2 ] 2 d A = ∬ R

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Author: Ron Larson; Bruce H. Edwards

Publisher: Cengage Limited

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Chapter 14.5, Problem 16E

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To calculate: The area of the surface given by z=f(x,y)=a2−x2−y2 which lies above the region R, represented by the limit shown R={(x,y): x2+y2≤a2}

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Chapter 14.5, Problem 15E

Chapter 14.5, Problem 17E

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The area of the surface given by z = f ( x , y ) = a 2 − x 2 − y 2 which lies above the region R , represented by the limit shown R = { ( x , y ) : x 2 + y 2 ≤ a 2 }

Solution Summary: The author calculates the surface area of the region R by using z=f(x,y)=sqrta2-x

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To calculate: The area of the surface given by z=f(x,y)=a2−x2−y2 which lies above the region R, represented by the limit shown R={(x,y): x2+y2≤a2}

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