   Chapter 14.5, Problem 25E

Chapter
Section
Textbook Problem

Finding Surface Area Using TechnologyIn Exercises 21–26, write a double integral that represents the surface area of z = f ( x , y ) that lies above the region R . Use a computer algebra system to evaluate the double integral. f ( x , y ) = 4 − x 2 − y 2 R = { ( x , y ) : 0 ≤ x ≤ 1 ,     0 ≤ y ≤ 1 }

To determine

To Calculate: A double integral that represents the surface area of z=f(x,y)=4x2y2 lying above the region R represented by R={(x,y): 0x1, 0y1} using a computer algebra system.

Explanation

Given: The surface is f(x,y)=4x2y2 lying above the region R represented by R={(x,y): 0x1, 0y1}

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

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

Calculation: The surface f(x,y)=4x2y2 above the region R={(x,y): 0x1, 0y1}.

First, find the partial differentiation of the surface with respect to x is found using ddx(xn)=nxn1,ddx(constant)=0.

fx(x,y)=ddx(4x2y2)=2x21+0=2x

Now, with respect to y

fy(x,y)=ddy(4x2y2)=02y2

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