Area of an Ellipse using Integral Calculus

Date: 11/4/96 at 0:24:13
From: Mrs. Allyne F. Dault
Subject: Area of an ellipse

I've been doing a research project on finding the area of an ellipse. 
I know the formula is 1/2 the length of the major axis times 1/2 the 
length of the minor axis times pi, but I want to know where it comes 
from.

Date: 11/4/96 at 8:22:29
From: Doctor Anthony
Subject: Re: Area of an ellipse

You need integral calculus to find the area of the ellipse.  If you 
think of the area in the first quadrant with x and y both positive, 
the area is given by INT(0 to a)[y dx]

Let the equation of the ellipse in parametric form be:

           x= a cos(theta)
           y= b sin(theta)

dx = -a sin(theta) d(theta) 

Area = INT(pi/2 to 0)[-b sin(theta) * a sin(theta) * d(theta)]
     = INT(0 to pi/2)[absin^2(theta) * d(theta)]
     = INT(0 to pi/2)[(ab/2)(1-cos(2theta) * d(theta)]
     = (ab/2)[(theta) - (1/2)sin(2theta)] from 0 to pi/2
     = (ab/2)[pi/2 - 0]
     = pi*ab/4

The total area of the ellipse will be 4 times this area, so:

Area of ellipse = pi*ab

-Doctor Anthony,  The Math Forum
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