   Chapter 15.3, Problem 22E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.22. Inside the sphere x2 + y2 + z2 = 16 and outside the cylinder x2 + y2 = 4

To determine

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

Explanation

Given:

The region D lies inside the sphere and outside the cylinder.

The spherical surface is, x2+y2+z2=16 .

The cylindrical region is,  x2+y2=4 .

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)

Calculation:

From the given region D, it is observed that the sphere and the cylinder intersects the xy-plane. So, the value of r varies from 2 to 4 and the value of θ varies from 0 to 2π .

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

DzdA=202π2416r2(r)drdθ=202π24r16r2drdθ

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

02π24r16r2drdθ=02πdθ24r16r2dr

Let t=r2

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