My Math Forum Solving cylinder with spherical coordinate triple integration

 Calculus Calculus Math Forum

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:

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

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
 shape.jpg (18.6 KB, 4 views)

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:
 Originally Posted by nietzsche 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.

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

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post ZMD Calculus 1 April 10th, 2017 04:39 AM chr0x Calculus 3 November 22nd, 2015 03:36 AM vinnyzwrx Calculus 1 August 3rd, 2013 02:41 PM cmmcnamara Calculus 1 January 14th, 2010 12:00 PM cmmcnamara Calculus 2 March 15th, 2009 05:51 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top