   Chapter 15.3, Problem 27E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Use the guidelines of Section 4.5 to sketch the curve. y=(x1)3x2

Single Variable Calculus: Early Transcendentals, Volume I

x2 2x 5 = 0

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Multiply: (3)(2)(4)(7)

Elementary Technical Mathematics

Which of the following areas are equal? Why?

Single Variable Calculus: Early Transcendentals

True or False: i + j is a unit vector in the direction of 5i + 5j.

Study Guide for Stewart's Multivariable Calculus, 8th

What is the integrating factor for xy′ + 6x2y = 10 − x3?

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 