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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding Surface AreaIn Exercises 3–16, find the area of the surface given by z = f ( x , y ) that lies above the region R .

f ( x , y ) = a 2 x 2 y 2 R = { ( x , y ) : x 2 + y 2 b 2 , 0 < b < a }

To determine

To calculate: The area of the surface given by z=f(x,y)=a2x2y2 which lies above the region R, represented by the limit shown R={(x,y): x2+y2b2, 0<b<a}

Explanation

Given: The surface is given by f(x,y)=a2x2y2, above the region R, represented by the limit shown R={(x,y): x2+y2b2, 0<b<a}

Formula used: The surface area can be calculated of the region R by,

S=R1+[fx(x,y)]2+[fy(x,y)]2dA

Differentiation formula ddx(xn)=nxn1,ddx(constant)=0

The equation of circle x2+y2=r2, where r is the radius

Calculation: The function given is f(x,y)=a2x2y2

Now partially differentiating it with respect to x, use ddx(xn)=nxn1,ddx(constant)=0

fx(x,y)=ddx(a2x2y2)=2x2a2x2y2=xa2x2y2

Now, with respect to y.

It is found that that:

fy(x,y)=ddx(a2x2y2)=2y2a2x2y2=ya2x2y2

Substitute in the formula S=R1+[fx(x,y)]2+[fy(x,y)]2dA

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Chapter 14 Solutions

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