On finding the circumference of an ellipse!

I've been looking through all the maths information I can find for a formula: the formula for ellipse circumference. Unluckily I can't find one, so I decided to create one with all the stuff I read.

In this Desmos link I have been able to figure out much of it, but I just can't solve the integral. Could you help me?
BTW the note mentioned in this link does not exist yet :)

#Geometry

Easy Math Editor

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Comments

@Zakir Husain

Jeff Giff - 1 month ago

The integrals involved in the circumference of an ellipse are non-elementary integrals (for e.g. Elliptic Integral of second kind )

These can either be approximated through series expansions or may be calculated directly by the computers.

For more info. you can view this video

Zakir Husain - 1 month ago

thanks sir!

Jeff Giff - 1 month ago

Hi sir I’ve got an integral here: $\int \sqrt{a^2\cos^2 \theta +b^2\sin^2\theta}~ d\theta$ Could you integrate it with respect to $\theta$ ?

Jeff Giff - 1 week, 6 days ago

$\int\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}d\theta$ $=|a|\int\sqrt{\cos^2\theta+\dfrac{b^2}{a^2}\sin^2\theta}d\theta$ $=|a|\int\sqrt{1-\sin^2\theta+\dfrac{b^2}{a^2}\sin^2\theta}d\theta$ $=|a|\red{\int\sqrt{1-\left(1-\dfrac{b^2}{a^2}\right)\sin^2\theta}d\theta}$ $=|a|\red{E( \theta | ( 1-\frac{b^2}{a^2}))}$ where $E(\theta | k)$ is elliptic integral of second kind

Zakir Husain - 1 week, 6 days ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$