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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.

13. The part of the surface z = 1/(1 + x2 + y2) that lies above the disk x2 + y2 ≤ 1

To determine

To find: The area of given surface correct to four decimal places by using integrating calculator.

Explanation

Given:

The function is the part of the surface z=11+x2+y2.

The region D lies above the disk x2+y21.

Formula used:

The surface area with equation z=f(x,y),(x,y)D, where fx and fy are continuous, is A(S)=D[fx(x,y)]2+[fy(x,y)]2+1dA.

Here, D is the given region.

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π, then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

Convert the given problems into the polar coordinates to make the problem easier. By the given the conditions, it is observed that, r varies from 0 to 1 and θ varies from 0 to 2π.

The partial derivatives fx and fy are,

fx=(1)(1+x2+y2)2(2x)=2x(1+x2+y2)2fy=(1)(1+x2+y2)2(2y)=2y(1+x2+y2)2

Then, by the equation (1), the area of surface is given by,

A(S)=

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Chapter 15 Solutions

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