Integration Limit problem!

Prove

$\displaystyle \int_0^{2\pi}{\sqrt{a^2\cos^2(t) + b^2 \sin^2(t)}\ dt} \geq \sqrt{4\pi (\pi a b + {(a-b)}^2)}$

As Paul J. Nahin describes this inequality considering perimeter of an ellipse using isoperimetric inequality and a nice trick, he also says that he has not been able to find a proof using integration manipulations.

I also tried it myself but I could not think of any method. Try it and post your solutions. I would be glad to see them. You can also send it to the author as he mentions in his book.

Easy Math Editor

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Comments

@Mark Hennings @Ishan Singh

Kartik Sharma - 4 years 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}`	$\boxed{123}$