×

# 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.

Note by Kartik Sharma
5 months, 1 week ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by: