Steinmetz SolidDate: 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/ |
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