Volume of an Elliptical ConeDate: 10/15/98 at 12:33:51 From: francesca Subject: Volume for elliptical cone Can you enlighten me on how to find the volume for an elliptical cone? I've tried many approximations. I'm looking for an equation or analysis by integration. Thanks, Francesca Date: 10/16/98 at 08:22:03 From: Doctor Rob Subject: Re: Volume for elliptical cone If it is a right elliptical cone, i.e., the elliptical base is perpendicular to the axis, then the formula is: V = Pi*a*b*h/3 where V is the volume, h is the height, and a and b are the semi-major and semi-minor axes of the elliptical base. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 10/16/98 at 15:39:45 From: francesca weaver Subject: Re: Volume for elliptical cone Thanks for the speedy reply. Could you show or explain, how you would begin to solve for the volume through integration (triple integral)? Just the initial setup - need not solve. Yesterday, I ended up solving for it through geometric properties, using an assumption that was valid in my case. I considered your exact equation by modifying that of the cone but had no proof/resource to check it against. Thanks much, Francesca Date: 10/16/98 at 16:11:50 From: Doctor Rob Subject: Re: Volume for elliptical cone The equation of the cone is x^2/a^2 + y^2/b^2 - z^2/h^2 = 0, for 0 <= z <= h, where the origin is at the vertex, the z-axis is the axis of the cone, the plane z = h is the plane of the base, the semi-major axis of the elliptical base is the line segment z = h, y = 0, 0 <= x <= a, and the semi-minor axis of the elliptical base is the line segment z = h, x = 0, 0 <= y <= b. The volume of the cone is then the triple integral: V = Int{Int[Int(1)dz]dy}dx where the limits of integration are: -a <= x <= a -b*sqrt(a^2-x^2)/a <= y <= b*sqrt(a^2-x^2)/a h*sqrt(x^2/a^2+y^2/b^2) <= z <= h The innermost integral can be done immediately. The remaining double integral can be simplified a little by using the symmetry of the ellipse, to: V = 4*h*Int{Int[1-sqrt(x^2/a^2+y^2/b^2)]dy}dx where the limits of integration are: 0 <= x <= a, 0 <= y <= b*sqrt(a^2-x^2)/a This is how I would set it up. Mathematica(TM) concurs that, when integrated, the answer is: V = Pi*a*b*h/3 - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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