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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

Use polar coordinates to find the volume of the given solid.

27. Inside both the cylinder x2 + y2 = 4 and the ellipsoid 4x2 + 4y2 + z2 = 64

To determine

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

Explanation

Given:

The region D lies inside the cylinders x2+y2=4 and the ellipsoid 4x2+4y2+z2=64 .

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:

Obtain the value for z from the ellipsoid.

4x2+4y2+z2=64z2=644x24y2z=±644x24y2z=±644(x2+y2)

And, the value of r varies from 0 to 2 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=02π02[644r2(644r2)](r)drdθ=2(2)02π02r16r2drdθ=402π02r16r2drdθ

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

402π02r16r2drdθ=402πdθ02r16r2dr

Let t=r2

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Chapter 15 Solutions

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