Area of Part of an Ellipse
Date: 04/07/2001 at 09:07:47 From: Phil Townsend Subject: Partial area of an ellipse Is there an equation to determine the area of part of an ellipse? Assume an ellipse that has a line bisecting it perpendicular to either the major or minor axis of the ellipse. What I want is the area of the ellipse either above or below that line, assuming that I know the distance (or height) to the line from the bottom of the ellipse. I know that this can be done for a circle, and I have searched for an equation for an ellipse to no avail.
Date: 04/09/2001 at 11:27:11 From: Doctor Rob Subject: Re: Partial area of an ellipse Thanks for writing to Ask Dr. Math, Phil. The formula you seek is the following. Suppose the major and minor axes of the ellipse have length 2*a and 2*b, and you seek the area between the tangent at one vertex and a parallel line a + d units from it, perpendicular to the major axis. Here -a <= d <= a. Then: A = a*b*(Pi/2 + Arcsin[d/a]) + b*d*sqrt(a^2-d^2)/a Notice that when d = a, and the parallel line is the tangent at the other vertex, you get A = a*b*Pi, the well-known formula for the area of the entire ellipse, and when d = 0, and the parallel line is the minor axis, you get A = a*b*Pi/2, half the area of the whole ellipse. To work with a tangent and a line both perpendicular to the minor axis, swap a and b in the above discussion. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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