I'm trying to learn this out of my own genuine curiosity.
If $f(x)$ gives you the rate of change, then $\displaystyle\int{f(x)}$ gives you area under the curve
How would I calculate the length of the curve though?
Lets say I have a graph of $x^2$, and I would like to get the distance the line travels between $x = -1$ and $x = 1$, how would I calculate that value?
The length $S$ of a curve $y(x) $in the interval $[a$, $b]$ is:
$$ S = \int ds = \int \sqrt{dx^2 + dy^2} = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx $$
Example. Take $a = -1$ and $b = 1$ and $y = x^2$. This means $\frac{dy}{dx} = 2x$ and so we get:
$$ S = \int \sqrt{1 + 4x^2} dx = \frac{x}{2} \sqrt{4x^2 + 1} + \frac{1}{4} \ln \vert 2x + \sqrt{4x^2 + 1} \vert + C$$
Taking the difference of above at $a = -1$ and $b = 1$ we get:
$$ S = \frac{1}{2} \sqrt{5} + \frac{1}{4} \ln (2 + \sqrt{5}) + \frac{1}{2} \sqrt{5} - \frac{1}{4} \ln (-2 + \sqrt{5}) $$
$$ S = \sqrt{5} + \frac{1}{4} \ln \left( \frac{2 + \sqrt{5}}{-2 + \sqrt{5}} \right) $$