   Chapter 15.3, Problem 25E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.25. Above the cone z = x 2 + y 2 and below the sphere x2 + y2 + z2 = 1

To determine

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

Explanation

Given:

The region D lies above the cone z=x2+y2 and below the sphere x2+y2+z2=1 .

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:

Substitute z=x2+y2 in x2+y2+z2=1 .

x2+y2+z2=1x2+y2+(x2+y2)2=12x2+2y2=1x2+y2=12

So, the value of r varies from 0 to 12 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=02π012(1r2r2)(r)drdθ=02π012(r1r2r(r))drdθ=02π012r1r2drdθ02π012r2drdθ

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

02π012r1r2drdθ02π012r2drdθ=02πdθ012r1r2dr02πdθ012r2dr

Let t=r2 .

Then, dt=2rdr .

Compute the volume as follows

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