Chapter 14.5, Problem 25E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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)=4âˆ’x2âˆ’y2 lying above the region R represented by R={(x,y):Â 0â‰¤xâ‰¤1,Â 0â‰¤yâ‰¤1}

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

S=âˆ¬R1+[fx(x,y)]2+[fy(x,y)]2dA

Calculation: The surface f(x,y)=4âˆ’x2âˆ’y2 above the region R={(x,y):Â 0â‰¤xâ‰¤1,Â 0â‰¤yâ‰¤1}.

First, find the partial differentiation of the surface with respect to x is found using ddx(xn)=nxnâˆ’1,ddx(constant)=0.

fx(x,y)=ddx(4âˆ’x2âˆ’y2)=âˆ’2x2âˆ’1+0=âˆ’2x

Now, with respect to y

fy(x,y)=ddy(4âˆ’x2âˆ’y2)=0âˆ’2y2âˆ’

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