Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Volume between three surfaces

  1. Jul 16, 2010 #1
    Dear all,

    I do really need your help.
    I'd like to find the volume contained between a sphere (x^2+y^2+z^2=r^2) , plane1 (ax+by+cz+d=0), and plane2 (z-h=0).

    Would you please check what I've done till now?

    From the sphere and plane1 equations I got:
    x1=sqrt*(r^2-y^2-z^2)
    x2=d/a-(b/a)y-(c/a)z

    Then, by assuming that x1=x2 (as the sphere and plane1 intersect), I derived two equations: one for y1 and one for y2 as follow:
    y1=f(z)
    y2=g(z)

    Also, by assuming y1=y2, I got:
    z1=M (a constant value)
    z2=N (a constant value)

    Now, let assume z1<h<z2

    So, to derive the volume I thought that I can run three integrals as bellow:

    A=int(1,x,x1..x2)
    B=int(A,y,y1..y2)
    volume = int(B,z,h..z2)

    Am I right? Would you please let me know that what I am doing is right or not?

    I also drew a simple picture. The shaded area represents the portion of the sphere that I am looking for its volume.

    Thanks in advance,
    Nejla
     

    Attached Files:

    Last edited: Jul 16, 2010
  2. jcsd
  3. Jul 18, 2010 #2
    Hi again,

    I tried it for a special case where plane1 is x=0 and I got the right answer. But for the general case where plane1 is ax+by+cz+d=0, I cannot calculate the integrals.

    For the general case I have:

    x1=- d/a - (b*y)/a - (c*z)/a

    x2=(r^2 - x^2 - y^2)^(1/2)

    y1=-(a*(a^2*r^2 - a^2*z^2 + b^2*r^2 - b^2*z^2 - c^2*z^2 - 2*c*d*z - d^2)^(1/2) + b*d + b*c*z)/(a^2 + b^2)

    y2=-(b*d - a*(a^2*r^2 - a^2*z^2 + b^2*r^2 - b^2*z^2 - c^2*z^2 - 2*c*d*z - d^2)^(1/2) + b*c*z)/(a^2 + b^2)

    z1=-(c*d + (a^4*r^2 + 2*a^2*b^2*r^2 + a^2*c^2*r^2 - a^2*d^2 + b^4*r^2 + b^2*c^2*r^2 - b^2*d^2)^(1/2))/(a^2 + b^2 + c^2)

    z2=-(c*d - (a^4*r^2 + 2*a^2*b^2*r^2 + a^2*c^2*r^2 - a^2*d^2 + b^4*r^2 + b^2*c^2*r^2 - b^2*d^2)^(1/2))/(a^2 + b^2 + c^2)

    I do really need your help.
    Thanks in advance,
    Nejla
     
  4. Jul 19, 2010 #3
    Thank you all. I eventually solved my problem. Whenever I get the chance, I will write down its answers here.


    Nejla
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Volume between three surfaces
  1. Help on volume (Replies: 1)

Loading...