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/