See also: answers with code on GameDev.SE
How can I find out the arc length of a Bézier curve? For instance, the arc length of a linear Bézier curve is simply:
$$s = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$$
But what of quadratic, cubic, or nth-degree Bézier curves?
$$\mathbf{B}(t) = \sum_{i=0}^n {n\choose i}(1-t)^{n-i}t^i\mathbf{P}_i$$