My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By Country Boy
Reply
 
LinkBack Thread Tools Display Modes
April 5th, 2018, 02:23 AM   #1
Newbie
 
Joined: Dec 2017
From: Netherlands

Posts: 21
Thanks: 0

Calculating Volume using Triple Integral

**I am trying to solve this problem but I am having difficulties to finish it. I would appreciate of someone can advice me on how to continue**

**Problem:**
Calculate
$$\iiint_{V} Z\mathrm dV$$
where E is defined by
$$ x^2+y^2 \le z^2 $$and$$ x^2+y^2=z^2 \le R^2 with R\gt0$$

**Solution**
Using Cylindrical Coordinates
$$\iiint_{V} Z\mathrm dV = \iiint_{S} Z\mathrm rdrdzd\theta $$
$$\iiint_{V} Z\mathrm dV = \iiint_{S} Z\mathrm rdrdzd\theta $$
$$x = rcos\theta, y= rsin\theta $$

Last edited by sosoebot; April 5th, 2018 at 02:30 AM.
sosoebot is offline  
 
April 5th, 2018, 03:47 AM   #2
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,264
Thanks: 902

What is "Z"? Do you mean just the z coordinate? What are "V" and "S"? You don't define them anywhere.
Country Boy is offline  
April 5th, 2018, 03:51 AM   #3
Newbie
 
Joined: Dec 2017
From: Netherlands

Posts: 21
Thanks: 0

Sorry V=E, and it is defined in the problem
sosoebot is offline  
April 5th, 2018, 11:58 PM   #4
Newbie
 
Joined: Dec 2017
From: Netherlands

Posts: 21
Thanks: 0

I have reposted the question here. I made alot of typo errors in the previous post

**I am trying to solve this problem but I am having difficulties to finish it. I would appreciate of someone can advice me on how to continue**

**Problem:**
Calculate
$$\iiint_{V} Z\mathrm dV$$
where V is defined by
$$ x^2+y^2 \le z^2 $$and$$ x^2+y^2+z^2 \le R^2 with R\gt0$$

**Solution**
Using Cylindrical Coordinates
$$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$
$$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$
$$x = rcos\theta, y= rsin\theta $$
sosoebot is offline  
April 6th, 2018, 04:02 PM   #5
Global Moderator
 
Joined: May 2007

Posts: 6,835
Thanks: 733

You still have one unanswered question. Does Z=z?
mathman is offline  
April 6th, 2018, 05:34 PM   #6
Newbie
 
Joined: Dec 2017
From: Netherlands

Posts: 21
Thanks: 0

Yes, Z=z as stated in the problem
sosoebot is offline  
April 6th, 2018, 08:32 PM   #7
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 2,304
Thanks: 961

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by sosoebot View Post
Yes, Z=z as stated in the problem
It does not state that anywhere in the problem.

Just remember, Mathematics is "case-sensitive."

Anyway, you have the bottom part of your post marked as "Solution." Do you need help in writing the limits on the integral? The non-trivial part of the limits is the restriction on the z coordinate. z would have to go from 0 to $\displaystyle \sqrt{x^2 + y^2}= \sqrt{R^2 - r^2}$ due to your inequality. Thus the integral would be
$\displaystyle \int _{0}^{2 \pi} \int _{0}^{R} \int _{0}^{\sqrt{R^2 - r^2}} z ~ r ~ dr ~ dz ~ d \theta$

-Dan

Last edited by topsquark; April 6th, 2018 at 08:40 PM.
topsquark is offline  
April 7th, 2018, 05:45 AM   #8
Newbie
 
Joined: Dec 2017
From: Netherlands

Posts: 21
Thanks: 0

Thank you very much. I have done the integration and I ended up with
$$\frac{\pi R^4}{4}$$
sosoebot is offline  
April 7th, 2018, 10:00 AM   #9
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,264
Thanks: 902

I don't see anywhere in the stated problem that z must be non-negative. The region of integration is that portion of the cone, $\displaystyle x^2+ y^2\le z^2$ that lies inside the sphere $\displaystyle x^2+ y^2+ z^2\le R^2$. Taking the square root ignores the lower nappe of the cone.

But since the cone is symmetric about the xy-plane, integrating z over both nappes gives a result, without the necessity of actually integrating, of 0.
Thanks from topsquark
Country Boy is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
calculating, integral, triple, volume



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Volume with Triple Integral sosoebot Calculus 2 April 3rd, 2018 12:43 PM
calculating volume of three intersecting cylinders zollen Calculus 1 October 8th, 2017 03:34 PM
Triple Integral of a volume of a pyramid zollen Calculus 8 May 2nd, 2017 02:38 PM
Double/Triple Integral for volume between surfaces zollen Calculus 8 April 9th, 2017 09:51 AM
volume of a sphere cut by a plane (triple integral) triplekite Calculus 1 December 5th, 2012 08:16 AM





Copyright © 2019 My Math Forum. All rights reserved.