Register to reply

Circumference of an ellipse

by StephenPrivitera
Tags: circumference, ellipse
Share this thread:
StephenPrivitera
#1
Jun17-03, 02:54 PM
P: 364
I have two integrals to give the circumference of an ellipse. I can't solve either.
First, using rectangular coordinates,
1/2s=S{[1+(f'(x))^2]^(1/2)}dx taken from x=-a to x=a
Since, y^2=b/a(a^2-x^2)
2y*y'=-2bx/a
y'=-bx/(ay)
[f'(x)]^2=(x^2)/(a^2-x^2)
At this point, I'm already uncomfortable because b is no longer in the equation, and clearly the circumference should depend on both a and b.
Next, using parametrics, I have
s=S[(bcosx)^2+(asinx)^2]^(1/2)dx from x=0 to x=2pi
This integral shows more promise for finding the answer. I expect the answer to be C=pi(a+b) simply because this would reduce to C=(2pi)r for the case when a=b. I've tried manipulating the second integral in every way possible to fit in trig substitution but it just won't work. It doesn't look like integration by parts will help. Of course, there's always the possiblity that these integrals do not give the circumference of an ellipse at all. Even so, it would be satisfying to find an answer.
Can someone give me a hint?
Phys.Org News Partner Mathematics news on Phys.org
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
chroot
#2
Jun17-03, 03:14 PM
Emeritus
Sci Advisor
PF Gold
chroot's Avatar
P: 10,427
It should not rightly be called the 'circumference,' which is a word reserved for circles. It is better to call it the 'perimeter.'

This is, in fact, a complicated topic. Here's a good resource to get you started:

http://home.att.net/~numericana/answer/ellipse.htm

- Warren


Register to reply

Related Discussions
Circumference of cirlce Precalculus Mathematics Homework 6
Proove its a circumference Precalculus Mathematics Homework 13
Circumference of an ellipse Calculus & Beyond Homework 7
Reduced Circumference Astronomy & Astrophysics 1
Circumference of an Ellipse Introductory Physics Homework 2