
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 9th, 2017, 01:08 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 175 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}^{2rsin(\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. Last edited by zollen; July 9th, 2017 at 01:12 PM. 
July 13th, 2017, 01:49 PM  #2 
Member Joined: Dec 2016 From:  Posts: 62 Thanks: 10 
Why is $\rho$ less than 2? the denominator is a sum but not of two positive terms necessarily.

July 13th, 2017, 02:31 PM  #3  
Senior Member Joined: Jan 2017 From: Toronto Posts: 175 Thanks: 2  Quote:
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.  

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 04:39 AM 
Finding cylinder mass using triple integrals  chr0x  Calculus  3  November 22nd, 2015 03:36 AM 
Triple integrals in spherical coordinates question  vinnyzwrx  Calculus  1  August 3rd, 2013 02:41 PM 
Spherical Coordinate Question  cmmcnamara  Calculus  1  January 14th, 2010 12:00 PM 
Spherical Coordinate Problem  cmmcnamara  Calculus  2  March 15th, 2009 05:51 PM 