   Chapter 15.5, Problem 10E

Chapter
Section
Textbook Problem

Find the area of the surface.10. The part of the sphere a x2 + y2 + z2 = 4 that lies above the plane z = 1

To determine

The area of given surface.

Explanation

Given:

The function x2+y2+z2=4.

The region D lies above the plane z=1.

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:

Solve the given equation of the function.

x2+y2+z2=4z2=4x2y2z=4x2y2

Convert the given problems into the polar coordinates to make the problem easier. To obtain the limits, solve the given equations as shown below.

x2+y2+12=4x2+y2=41x2+y2=3

By given the conditions, it is observed that, r varies from 0 to 3 and θ  varies from 0 to 2π.

Obtain the partial derivatives of f with respect to x and y.

fx=12(4x2y2)12(2x)=x(4x2y2)12fy=12(4x2y2)12(2y)=y(4x2y2)12

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

A(S)=D(x(4x2y2)

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