# Euler Equation to Compute Extreme?

1. Oct 24, 2012

### black_hole

A problem on my homework:

We learn early on that "the shortest distance between two points is a straight line." Let's prove it...Using the Euler equation, compute the extrema of

∫sqrt(1 + (dy/dx)2)dx from x1 to x2 ...show that this corresponds to lines "y = mx +b".

Euler had a lot of different equations. I recognize this as the formula for calculating the length of a graph but that's not what I'm being asked to do. Does anyone what formula this is referring to?

Thanks!

2. Oct 24, 2012

### Hurkyl

Staff Emeritus
If the hint is confusing, then ignore it for the moment. Amongst all paths from one point to another, how would you go about determining which ones have the shortest length?

3. Oct 24, 2012

### HallsofIvy

Staff Emeritus
I would presume that they are refering to the "Euler equation" (also known as the "Euler-Lagrange equation") for "Calculus of variations" since that is the kind of problem Calculus of variations solves.

http://mathworld.wolfram.com/CalculusofVariations.html

4. Oct 26, 2012

### black_hole

Thanks HallsofIvy. That was exactly what I was looking for!