   Chapter 15.3, Problem 23E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.23. A sphere of radius a

To determine

To find: The volume of the given solid by using polar coordinates.

Explanation

Given:

The region D is a sphere of radius a.

Formula used:

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)

aaf(x)dx=20af(x)dx (3)

Calculation:

From the given region D, let the equation of the sphere be x2+y2+z2=a2 .

Therefore, z=a2x2y2 .

Thus, r varies from -a to a and θ varies from 0 to 2π .

Substitute x=rcosθ and y=rsinθ in the equation (1) and obtain the required volume.

DzdA=02πaaa2r2(r)drdθ=02πaara2r2drdθ

Integrate the function with respect to r and θ by using the equation (2).

02πaara2r2drdθ=02πdθaara2r2dr=202πdθ0ara2r2dr           [by equation (3)]

Let t=r2

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