My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
July 9th, 2017, 02:08 PM   #1
Senior Member
 
Joined: Jan 2017
From: Toronto

Posts: 152
Thanks: 2

Solving cylinder with spherical coordinate triple integration

Consider the region R within the cylinder x^2 + y^2 <= 4, bounded below by z = 0 and above by z = 2 - y. Assume a mass density = z.
Set up and evaluate the integral representing the mass of the solid.

This is easy with cylinderical coordinates:

$\displaystyle
\int_{0}^{ 2\pi} \int_{0}^{2} \int_{0}^{2-rsin(\theta)} z r dz dr d \theta
$

However I would like to solve this problem by using sphereical coordinates:

$\displaystyle
\int_{0}^{2\pi} \int_{0}^{\pi/2} \int_{2}^{???} \rho cos(\phi) \rho^2 sin(\phi)
$

To solve the upper limit of ???:

$\displaystyle
z = 2 - r sin(\theta) ==> \rho cos(\phi) = 2 - \rho sin(\phi) sin(\theta)
, \rho = 2 / (cos(\phi) + sin(\phi) sin(\theta))
$

However $\displaystyle \rho$ would be less than 2, which does not make sense because I expected $\displaystyle \rho$ should be at least 2 or more

Any tips would be much appreciated. Thanks.
Attached Images
File Type: jpg shape.jpg (18.6 KB, 4 views)

Last edited by zollen; July 9th, 2017 at 02:12 PM.
zollen is offline  
 
July 13th, 2017, 02:49 PM   #2
Member
 
Joined: Dec 2016
From: -

Posts: 54
Thanks: 10

Why is $\rho$ less than 2? the denominator is a sum but not of two positive terms necessarily.
nietzsche is offline  
July 13th, 2017, 03:31 PM   #3
Senior Member
 
Joined: Jan 2017
From: Toronto

Posts: 152
Thanks: 2

Quote:
Originally Posted by nietzsche View Post
Why is $\rho$ less than 2? the denominator is a sum but not of two positive terms necessarily.
$\displaystyle
2/(cos(\phi) + sin(\phi)cos(\theta)) <= 2
$

I would not be able to integrate with this upper limit since it is possible for the upper limit to be less than the lower limit.
zollen is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
coordinate, cylinder, integration, solving, spherical, triple



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Spherical Coordinate Map ZMD Calculus 1 April 10th, 2017 05:39 AM
Finding cylinder mass using triple integrals chr0x Calculus 3 November 22nd, 2015 04:36 AM
Triple integrals in spherical coordinates question vinnyzwrx Calculus 1 August 3rd, 2013 03:41 PM
Spherical Coordinate Question cmmcnamara Calculus 1 January 14th, 2010 01:00 PM
Spherical Coordinate Problem cmmcnamara Calculus 2 March 15th, 2009 06:51 PM





Copyright © 2017 My Math Forum. All rights reserved.