Spot the error (circumference of ellipse)

In summary, the conversation discusses calculating the circumference of an ellipse using the semi-major and semi-minor axes. The proof involves using the dirac delta function and the final answer is 4π. The conversation ends with a humorous mention of moving the discussion to a math forum where the use of dirac delta may cause some controversy.
  • #1
nonequilibrium
1,439
2
Spot the error:

The circumference of an ellipse with semi-major axis = 2 and semi-minor axis = 1 (i.e. [itex] \left( \frac{x}{2} \right)^2 + y^2 = 1 [/itex]) is [itex]4 \pi[/itex]. Proof:

[itex]\textrm{circumference} = \iint_{\mathbb R^2}{ \delta \left( \left( \frac{x}{2} \right)^2 + y^2 - 1 \right) } \mathrm d x \mathrm d y = 2 \iint_{\mathbb R^2}{ \delta \left( \left( \frac{x}{2} \right)^2 + y^2 - 1 \right) } \mathrm d \frac{x}{2} \mathrm d y = 2 \iint_{\mathbb R^2}{ \delta \left( x^2 + y^2 - 1 \right) } \mathrm d x \mathrm d y = 2 \times 2 \pi = 4 \pi [/itex]

(I couldn't post this in the Math forum since they would just yell at me for using the dirac delta, and it's also not homework as I know the answer.)
 
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  • #2
I could not see the mistake yet.. Interesting way to calculate the circumference :)
 
  • #3
mr. vodka said:
(I couldn't post this in the Math forum since they would just yell at me for using the dirac delta

Well, I just moved it to the math forum. So prepare for some yelling :biggrin:
 

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

The formula for calculating the circumference of an ellipse is 2π√((a^2 + b^2)/2), where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

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

No, the formula for the circumference of a circle (2πr) cannot be used to calculate the circumference of an ellipse because the shape of an ellipse is more elongated and does not have a constant radius.

3. How accurate is the formula for calculating the circumference of an ellipse?

The formula for calculating the circumference of an ellipse is an approximation and may not be 100% accurate. However, as the ellipse becomes more circular (a and b approach the same value), the accuracy of the formula increases.

4. Are there any other methods for calculating the circumference of an ellipse?

Yes, there are other methods such as using numerical integration or series approximation. These methods may provide more accurate results but can be more complicated to use.

5. Can the circumference of an ellipse be greater than the circumference of a circle with the same major axis?

Yes, the circumference of an ellipse can be greater than the circumference of a circle with the same major axis. This is because the ellipse has a longer path around its perimeter due to the elongation of its shape.

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