How can I use integrals to find the circumference of an ellipse?

[StephenPrivitera]

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?

 

[chroot]

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

 

[tacman]

It looks like you're on the right track with your second integral using parametric equations. However, there are a few things to note. First, the parameter x should range from 0 to 2pi, not -a to a. Also, the square root term should be squared before taking the integral, giving you:

s = S[(bcosx)^2 + (asinx)^2]dx from x=0 to x=2pi

= S[b^2cos^2x + a^2sin^2x]dx from x=0 to x=2pi

= b^2S[cos^2x]dx + a^2S[sin^2x]dx from x=0 to x=2pi

= b^2S[1/2(1+cos2x)]dx + a^2S[1/2(1-cos2x)]dx from x=0 to x=2pi

= b^2/2Sdx + a^2/2Sdx from x=0 to x=2pi

= b^2pi + a^2pi

= pi(a^2+b^2)

This is the correct formula for the circumference of an ellipse. It's important to note that this formula is dependent on both a and b, as you correctly suspected. So, the final answer is not C=pi(a+b), but rather C=pi(a^2+b^2). Keep in mind that the circumference of an ellipse is not simply the sum of the major and minor axes, as it would be for a circle. This is because the shape of an ellipse is not symmetrical like a circle, so the distance around the perimeter is greater. I hope this helps you solve the problem!