Explanation Given: The surface is f ( x , y ) = e − x sin y lying above the region R represented by, R = { ( x , y ) : x 2 + y 2 ≤ 4 } . 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 Calculation: The surface f ( x , y ) = cos ( x 2 + y 2 ) above the region R = { ( x , y ) : x 2 + y 2 ≤ π 2 } . First, find the partial differentiation of the surface with respect to x with the use of, d d x ( x n ) = n x n − 1 , d d x ( constant ) = 0 , d d x ( sin x ) = cos x . f x ( x , y ) = d d x ( cos ( x 2 + y 2 ) ) = − sin ( x 2 + y 2 ) ⋅ ( 2 x ) = − 2 x sin ( x 2 + y 2 ) Now, with respect to y . To get, f y ( x , y ) = d d y ( cos ( x 2 + y 2 ) ) = − s i n ( x 2 + y 2 ) ⋅ ( 2 y ) = − 2 y sin ( x 2 + y 2 ) As the region provided is R = { ( x , y ) : x 2 + y 2 ≤ π 2 }

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14.5, Problem 28E

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To calculate: A double integral that represents the surface area of z=f(x,y) lying above the region R={(x,y): x2+y2≤4}, where surface is f(x,y)=e−xsiny

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Chapter 14.5, Problem 27E

Chapter 14.5, Problem 29E

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Chapter 14 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrd Ch. 14.1 - Prob. 1E Ch. 14.1 - Evaluate the integral: xx2yxdy Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Evaluate the integral: 04x2x2ydy Ch. 14.1 - Evaluate the integral: x3x(x2+3y2)dy Ch. 14.1 - Evaluate the integral: eyyylnxxdx;y0 Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dx Ch. 14.1 - Evaluate the integral: 0x2yeyxdy

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A double integral that represents the surface area of z = f ( x , y ) lying above the region R = { ( x , y ) : x 2 + y 2 ≤ 4 } , where surface is f ( x , y ) = e − x sin y

Solution Summary: The author calculates a double integral that represents the surface area of f(x,y)=e-xmathrmsiny.

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To calculate: A double integral that represents the surface area of z=f(x,y) lying above the region R={(x,y): x2+y2≤4}, where surface is f(x,y)=e−xsiny

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Chapter 14 Solutions