Results 1 to 7 of 7

Math Help - Find the volume of a solid bound by two spheres and cone

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    4

    Find the volume of a solid bound by two spheres and cone

    I am trying to find the volume of intersection of a cone with two spheres:

     x^2+y^2+z^2=R_1^2

     x^2+y^2+(z-z_0)^2=R_2^2

     z=\sqrt{\frac{x^2+y^2}{r}}

    Can anyone help me out?

    Thank you!

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    I'm not clear on what you mean by "the volume bounded by". What volume is bounded and whether or not there is any volume bounded by these figures depends upon specific values. One sphere has center at the origin and the other has center at (0, 0, z_0). Whether there is any volume bounded by them and the cone depends upon the relation between R_1 and R_2, the radii of the two spheres, as well as the relation between them and z_0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    4
    Of course, if there are no solid bordered with these three surfaces, then the volume of intersection will be zero.
    But, if we assume that there is such a volume that is greater than zero, what will be the equation of it? What are the limits of integration?


    Tnx
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Quote Originally Posted by andra7 View Post
    I am trying to find the volume of intersection of a cone with two spheres:

     x^2+y^2+z^2=R_1^2

     x^2+y^2+(z-z_0)^2=R_2^2

     z=\sqrt{\frac{x^2+y^2}{r}}

    Can anyone help me out?

    Thank you!

    Right off the bat I'd plug numbers in it and see what's going on. Here's three:

    x^2+y^2+z^2=1

    x^2+y^2+(z-1)^2=2

    z=\sqrt{\frac{x^2+y^2}{2}}

    The plot below is a cross-section of the three along the y-plane. From this, I'd assume you mean the volume element in purple. Then how about using volume by discs:

    \mathop\int\limits_{\hspace{-6pt}\text{Purple}} \pi r^2 dz=\mathop\int\limits_{p_1}+\mathop\int\limits_{p_  2}

    . . . or maybe it's some other piece in there that you want.
    Attached Thumbnails Attached Thumbnails Find the volume of a solid bound by two spheres and cone-myvolumeplot.jpg  
    Last edited by shawsend; January 4th 2010 at 07:40 AM. Reason: changed dy to dz
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2010
    Posts
    4
    Actually, I mean the volume element in red on the plot below.
    Attached Thumbnails Attached Thumbnails Find the volume of a solid bound by two spheres and cone-slika.jpg  
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Very good then. Do a few with real numbers, then figure out how to generalize it to arbitrary constants. So, let's look first at the set:

    x^2+y^2+z^2=1

    x^2+y^2+(z-2)^2=2

    z=\sqrt{\frac{x^2+y^2}{1/10}}

    the cross-section of which is shown in the first plot. And I guess if my life dependent on it, I could figure out how to color in that section you want but it's not easy for me so you know which part we're lookin' at. Ain't that still circles in there because of the cone? Then can't I still use as a first effort, the disc metod:

    V=\mathop\int\limits_{\hspace{-8pt}\substack{\text{low}\\ \text{section}}} \pi r^2 dz+\mathop\int\limits_{\hspace{-8pt}\substack{\text{mid}\\ \text{section}}} \pi r^2 dz+\mathop\int\limits_{\hspace{-8pt}\substack{\text{top}\\ \text{section}}} \pi r^2 dz

    where I've drawn horizontal lines to separate the low, mid, and top sections which I've tried to show in the second plot but still hard to see the small low section. You know what I mean though.
    Attached Thumbnails Attached Thumbnails Find the volume of a solid bound by two spheres and cone-volumea.jpg   Find the volume of a solid bound by two spheres and cone-volumeb.jpg  
    Last edited by shawsend; January 4th 2010 at 07:39 AM. Reason: changed dy to dz
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jan 2010
    Posts
    4
    I think I can figure out from here.
    Thanks so much!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 17th 2011, 09:40 PM
  2. Find Volume of Cone
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: September 7th 2010, 08:15 AM
  3. Find volume of a solid bound by 2 functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 10th 2009, 09:13 AM
  4. Replies: 2
    Last Post: May 18th 2008, 04:45 AM
  5. Volume of solid inside 2 spheres
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 25th 2007, 01:17 PM

Search Tags


/mathhelpforum @mathhelpforum