Question

Line integral guide-thingy

Answer 1

I'm gonna start off without vectors, I'm going to derive the arclength formula. So we have f(x) and want to find the length. We can chop it up into a bunch of pieces that are like little hypotnuses (hypotni? lol) and use the pythagorean theorem to represent just the general formula for any one of these little pieces.

So we can see this relationship is true\[\Large (ds)^2 = (dx)^2+(dy)^2\] Now let's find the general formula for ds, which is just a piece of f(x), See how we're taking a geometric object and breaking it down into parts? \[\Large ds = \sqrt{(dx)^2+(dy)^2}\] Alright now what's the arc length? It's actually pretty simple. We just need to pick two points on the curve f(x) and then add up all the pieces from one end to the other. That's just the arc length, so we luckily have something to add up an infinite number of infinitely small things, and the integral is to the rescue...