Steinmetz Solid

Date: 06/08/2003 at 10:42:09
From: Jude
Subject: Volumes

Two pipes (radius = r) are crossing each other normally. What is the 
common part volume?

I can't make equations which I could integrate (normal functions).

Date: 06/13/2003 at 17:20:53
From: Doctor Douglas
Subject: Re: Volumes

Hi Jude,

Thanks for writing to the Math Forum.

If by "normally" you mean that the axes of the two cylinders intersect 
at right angles, then the common part is sometimes known as the 
"Steinmetz solid" or a "bicylinder."

To find the volume of this object, suppose that the cylinders each 
have radius r and that their axes extend along the x- and z-axes. Then 
the equations of the two cylinders are

  x^2 + y^2 = r^2             (z-axis cylinder);
  y^2 + z^2 = r^2             (x-axis cylinder).
   
Now imagine slicing the volume with planes that are perpendicular to 
the y-axis. If you look down the y-axis toward the origin, each of the 
slices of the solid has a square cross section. And the half-length of 
each side of the square is s(y) = sqrt(r^2-y^2). Hence the volume of 
each square slice is

    dV = dy*[2s(y)]^2 = 4*(r^2-y^2)*dy

Now all we have to do is to integrate this function for values of y 
between its smallest and largest values, i.e., -r < y < r:

    V = Int{-r<y<r} dV
      = Int{-r<y<r} 4*(r^2 - y^2)dy          
      = 8*(r^2*y - y^3/3) |{0<y<r}           integrand is even
      = 8*(r^3 - r^3/3)
      = (16/3)r^3

For more information about the bicylinder and related shapes, check 
out the following web page:

   Steinmetz Solid - Eric Weisstein's World of Mathematics
   http://mathworld.wolfram.com/SteinmetzSolid.html 

In addition to the simple calculus derivation above, it also outlines
the calculation of the volume by setting up a triple integral in
the coordinates {x,y,z} with the correct limits of integration.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/