How do you find the length of a plot?

Question

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.

Answer 1

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.

Answer 2

First, note the difference between a 'line', which is typically straight, and a 'curve', which can effectively be any continuous shape. The length of a line segment is easily expressed by the Euclidean distance between the two points. On a 2d plane, this is simply $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. For a curve, the length is called the arc length, which has numerous formulae, depending on how the curve is expressed. As you correctly surmised, the arc length can be found by summing infinitesimal line segments (i.e. integration) so this is in the realm of calculus. The arc length of a curve expressed by $y=f(x)$, between $x=a$ and $x=b$ is $\int_a^b\sqrt{1+f'(x)^2}\ \mathrm{d}x$ but for many curves, the arc length has no closed form solution.