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

In summary, the conversation discusses two different methods for finding the perimeter of an ellipse - one using rectangular coordinates and the other using parametrics. The first method involves an integral that becomes complicated when b is eliminated from the equation. The second method shows more promise, but the integrand does not seem to fit into any known techniques. It is also mentioned that the term "circumference" may not be appropriate for ellipses and a resource is provided for further information on the topic.
  • #1
StephenPrivitera
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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?
 
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  • #2
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 [Broken]

- Warren
 
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  • #3


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!
 

1. What is the formula for finding the circumference of an ellipse?

The formula for finding the circumference of an ellipse is C = π * (3/2 * (a + b) - √(a * b)) where a and b are the lengths of the semi-major and semi-minor axes of the ellipse.

2. Can the circumference of an ellipse be calculated using the same formula as a circle?

No, the formula for finding the circumference of a circle (C = 2πr) only applies to a perfect circle. The formula for an ellipse takes into account the unequal lengths of its two axes.

3. How does the eccentricity of an ellipse affect its circumference?

The eccentricity of an ellipse (e) is the measure of how "squished" or elongated the ellipse is. The closer e is to 1, the more elongated the ellipse is. As e approaches 1, the circumference of the ellipse approaches the circumference of a circle with the same diameter.

4. Can the circumference of an ellipse be greater than the circumference of a circle with the same diameter?

Yes, the circumference of an ellipse can be greater than the circumference of a circle with the same diameter. This is because the formula for the circumference of an ellipse takes into account the unequal lengths of its two axes, while the formula for a circle assumes equal lengths for the radius.

5. Are there any real-life applications for calculating the circumference of an ellipse?

Yes, there are many real-life applications for calculating the circumference of an ellipse. For example, it is used in engineering and architecture to determine the circumference of curved structures like bridges, arches, and domes. It is also used in astronomy to calculate the orbits of planets and other celestial bodies, which often follow an elliptical path.

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