Results 1 to 8 of 8

Math Help - Finding volume by using triple integral

  1. #1
    Junior Member
    Joined
    Feb 2011
    Posts
    30

    Finding volume by using triple integral

    I need to find the volume enclosed by x^2+y^2+z^2=a^2 and x^2+y^2=ax where a>0.

    How do I find the bounds? Do I apply spherical coordinates as written?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,560
    Thanks
    1425
    To start with, I expect that you have written the second bound wrongly, since that is the equation of a plane figure, not a solid...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2011
    Posts
    30
    I edited the first equation, but what's wrong with the second one?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,560
    Thanks
    1425
    It's a 2 Dimensional object, i.e. a plane figure. How are we supposed to know where along the \displaystyle z axis it's supposed to lie?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2008
    Posts
    1,034
    Thanks
    49
    Assuming the cylinder extends indefinitely up and down the z dimension, we have Viviani's Curve. Doing just the top half, we have z going from 0 (where it 'starts', on the (x,y) plane) up to \sqrt{a^2 - r^2} (where it hits the hemisphere). And we have r going from 0 at the centre (z axis), up to a cos theta i.e. everywhere inside the cylinder. And theta is turning through the x-positive half of the space, i.e. from minus pi/2 to pi/2. So...

    \displaystyle{V = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\ \int_{0}^{a \cos \theta} \int_{0}^{\sqrt{a^2 - r^2}} r\ dz\ dr\ d\theta}

    Just in case a picture helps to follow through from the inside out, we can start bottom left here, integrating r with respect to z...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case r), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    The general drift is...


    Which leaves a couple of blanks to fill. Hope this helps.

    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; March 12th 2011 at 04:22 AM. Reason: url's
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    Quote Originally Posted by Prove It View Post
    To start with, I expect that you have written the second bound wrongly, since that is the equation of a plane figure, not a solid...
    No, it's not. x^2+ y^2= ax where z can be anything is a cylinder. Specifically, it is the cylinder with central axis (a/2, 0, z) and radius a/2.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,560
    Thanks
    1425
    Quote Originally Posted by HallsofIvy View Post
    No, it's not. x^2+ y^2= ax where z can be anything is a cylinder. Specifically, it is the cylinder with central axis (a/2, 0, z) and radius a/2.
    Then it should have been stated that \displaystyle z can be anything...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    No, it was stated that this problem was in three dimensions. The fact that there was no restriction put on z meant that it could be anything.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Triple integral volume
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 6th 2011, 10:01 AM
  2. Replies: 0
    Last Post: November 14th 2010, 06:01 PM
  3. volume, triple integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 17th 2010, 08:09 PM
  4. triple integral - volume
    Posted in the Calculus Forum
    Replies: 6
    Last Post: November 29th 2009, 08:03 AM
  5. Triple integral - volume
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 14th 2008, 10:39 PM

Search Tags


/mathhelpforum @mathhelpforum