u/ElegantPoet3386

So random thought occured to me in math class and I want to know if my idea makes sense

So, most people know integrals as just area under the curve, or the antiderivative of a function, but really, it's just about summing a bunch of small things up. With that in mind, let's say we have a curve on the interval [a,b], and we want to find its exact length.

My idea is, draw a secant line segment connecting the points at [a,b]. It's going to be a pretty bad approximation obviously. But, what if we try drawing 2 secant lines segments, 1 bounded by [a,(a+b)/2] and the second bounded by [(a+b)/2, b]? Now the approximation is still bad, but it should be a bit better. Well, what if we try drawing 4 segments? Or 8? The approximation should be getting better and better.

Now, here's the part I'm a little unsure of. If we were to draw a near infinite amount of secant segments, would the sum of all the lengths of the secant segments approach the exact length of the curve? This is what I have in mind right now.

https://imgur.com/a/Ikhwvhg

Assuming what I'm unsure of is true, and, with what I said earlier about an integral just summing up a bunch of small things, if we take the limit as the number of segments approaches infinity, we should get the integral from a to b of the length of each segment dx equals the length of the curve.

As for getting that length, one way to find the length of a segment is to consider it the hypotenuse of a right triangle. To find the hypotenuse of this triangle, you can just use the triangle theroum thingy I forgot the name of where a^2 + b^2 = c^2.

In this case, a would be Δx, and b would be Δy. so the length of the hypotenuse would be sqrt(Δx^2 + Δy^2). And of course as the amount of segments approaches infinity, Δx becomes dx, and Δy becomes dy.

So, my theroetical method to calculate the length of a curve would just be the definite integral from a to b of sqrt(dx^2 + dy^2) dx. I'm not sure how would you find dx and dy, but if you could, and assuming all my logic has been correct, this should be the formula for the length of a curve.

So the question now is, is any of this correct?