Chapter 14.5, Problem 16E

### 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 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 ≤ a 2 }

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+y2a2}

Explanation

Given: The surface is given by f(x,y)=a2âˆ’x2âˆ’y2, above the region R, represented by the limit shown R={(x,y):Â x2+y2â‰¤a2}

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

S=âˆ¬R1+[fx(x,y)]2+[fy(x,y)]2dA

Differentiation formula ddx(xn)=nxnâˆ’1,ddx(constant)=0

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

Calculation: The function given is f(x,y)=a2âˆ’x2âˆ’y2.

Now partially differentiating it with respect to x, use ddx(xn)=nxnâˆ’1,ddx(constant)=0

fx(x,y)=ddx(a2âˆ’x2âˆ’y2)=âˆ’2x2a2âˆ’x2âˆ’y2=âˆ’xa2âˆ’x2âˆ’y2

Now, with respect to y

It is found that:

fy(x,y)=ddx(a2âˆ’x2âˆ’y2)=âˆ’2y2a2âˆ’x2âˆ’y2=âˆ’ya2âˆ’x2âˆ’y2

Putting in formula

S=âˆ¬R1+[âˆ’xa2âˆ’x2âˆ’y2]2+[âˆ’ya2âˆ’x2âˆ’y2]2dA=âˆ¬R

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