Area of an Ellipse using Integral CalculusDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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