   Chapter 14.5, Problem 15E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 7-12, solve for y in terms of x. 2x+y6=11

Calculus: An Applied Approach (MindTap Course List)

Expand each expression in Exercises 122. (2xy)y

Finite Mathematics and Applied Calculus (MindTap Course List)

Prove the statement using the , definition of a limit. limx12+4x3=2

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 69-74, rationalize the numerator. 70. x324

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach 