Alright I'll try to explain it to you. If I can't sufficiently, sorry, and if I explain what you already know, sorry about that too.can someone explain to me how the distance formula is related to the arc length formula?
my textbook has something demonstrating how they used the arc length formula to find the distance between (X1, Y1) and (X2, Y2), but i don't really get why.
If you know what the pythagorean theorem is, the distance formula should be simple. Imagine two points on a graph.
So the distance between them must be:
The distance formula (which I assume you know) uses the pythagorean theorem and uses the fact that you know both shorter legs of the triangle (as long as you know the coordinated of each point).
From here it's just c^2 = a^ + b^2.
And of course, c=d, a=x2-x1 (or y2-y1), b=y2-y1 (or x2-x1). Easy.
Now to understand integration:
In calculus (at least at the level you'll be dealing with your whole life presumably, lol), an integral is a limit of sums. Say you have this curve.
Now you know s (arc length) = S (a to b) sqrt( 1 + f'(x)^) dx
Ok, so now on differentials: differentials (e.g. dx, dy, dA, whatever) represent tiny changes in a variable. dx is a tiny change in x, dy a tiny change in y. The slope of a graph at a point = rise over run, or dy/dx, which is vertical change over horizontal change. It's important for you to know that a differential is a tiny change in direction.
Now look at that little dx hanging outside the sqrt( 1 + f'(x)^). distribute him into the equation.
s = S (a to b) sqrt (dx^2 + dy^2)
Now its starting to look like the distance formula.
All you have to know now is that the definition of an integral is a limit of sums.
If you think about it, dx and dy are tiny distances in the x and y direction, and so are the (x2-x1) and (y2-y1). The key difference here is that when you integrate, you take the sum of each of these differentials as dx and dy become infinitely small.
If your dx and dy are larger, you'll have a graph like this:
As your dx and dy become smaller, you get graphs that look like this:
And then
As you can see, you are doing the distance formula several times: the more times you do it (and the smaller the dx and dy), the more accurate your answer will be.
So the formula for arc length is just a sum of the results of the distance formula, as your dxs and dys become infinitely tiny and you use more and more pieces. The approximation becomes better and better.
I hope that explained your question. If not I just wasted a whole lot of time doing nothing