To start with, what is the length of a line from x=a to x=b? If you were to take a piece of string, and lay it across the plot of the line from a to b, how long would it be? For a line, the length from a to b is the same as the length from 0 to b-a. Which is just the length from 0 to 1, multiplied by b-a. Note that these lines are a special case, but I will talk about that later. So in the case of of a line mx+c, the length is sqrt(1+m^2)*(b-a). So that’s a line segment, but the real problem is finding a way to do this for any function, like x^2. If you are given a lower bound a, an upper bound b, and a polynomial c+dx+ex^2+fx^3… can you find the length of the plot of that polynomial from a to b. You can approximate any function with a bunch of line segments. If add more line segments it is a better and better approximation, which is why I am tagging this as calculus.
How do you find the length of a plot
analytic geometryarc lengthcalculus
Related Solutions
Your intuition is on point. I do not know your math background, but it sounds like you are at least familiar with basic calculus. The name for the line which you describe is arc length. We can pretty much approximate the arc length of any function, and obtain the exact value for quite a few types of functions. There are some pathological cases for which we cannot find exact values once we get into more advance stuff . For example, in general it is very difficult to obtain the arc lenght of a Fourier series restricted to a suitable domain.
Here is an edited version of the explanation of the arc length for single variable functions. This is directly from James Stewarts Calculus book, page 540, $7^{th}$ Edition. The picture here makes a little more precise your intuition, I hope this helps.
To answer your specific question about $f(x)=x^2.$
$$L=\int_ a^b \sqrt{1+(2x)^2}dx$$ $$=\frac{1}{2} x \sqrt{1+4 x^2}+\frac{1}{4} \text{ArcSinh}(2 x)$$ $$ = \frac{1}{2} x \sqrt{1+4 x^2}+\frac{1}{4} \log \left(\left|2x+\sqrt{(2x)^2+1} \right|\right)$$
So if you want to know the length of "the line" as you refer to it, just choose the interval $[a,b]$ for which you want to know and plug in the endpoint values accordingly. "Log" has has base $e.$
It's somewhat simpler, I think, to characterize maps that don't increase length rather than those that preserve it.
A map $f: X \to Y$ (where $X$ and $Y$ are metric spaces, with metric denoted $d$ in both cases) is said to be contractive if $d(f(x),f(y)) \le d(x,y)$ for all $x, y \in X$.
EDIT (following Jim Belk's remark) The length of a curve $C$ is the infimum of $L$ such that there exists a contractive map from $[0,L]$ onto $C$.
Best Answer
For a curve described by an equation
$$y=f(x),$$ the arc length is given by the integral
$$\int_a^b\sqrt{1+f'^2(x)}\,dx$$
(hoping you've heard of integrals).
Unfortunately, such an integral rarely has a closed-form expression. In the case of a quadratic polynomial $y=px^2+qx+r$,
$$\int_a^b\sqrt{1+(2px+q)^2}\,dx=\left.\frac{(q+ 2px) \sqrt{1 + (q+ 2px)^2} + \text{arsinh}(q+ 2 px)}{4p}\right|_a^b.$$
For higher degrees, you need special functions, such as Elliptic integrals, or numerical methods, such as Simpson's rule.