My course notes define a point between two control points along a Catmull Rom spline as a function of a parameter $u$, where
$$f(u) = \begin{bmatrix} u^3 & u^2 & u & 1 \end{bmatrix} \begin{bmatrix} -s & 2 - s & s - 2 & s \\ 2s & s - 3 & 3 - 2s & -s \\ -s & 0 & s & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{bmatrix}$$
From the section Wikipedia article on curvature pertaining to a graph in a plane, I've come up with the following integral:
$$\int_{a}^{b} {|f''(u)|\over(1 + f'(u)^2)^{3/2}} \space du$$
Where $a$ and $b$ are values of $u$ between $0$ and $1$, eg. a segment of the spline between two control points.
The only thing is, I have no idea how to go about getting from the above integral to a function $c(a,b) = $ the total of the curvature over said segment.
Have I made this too complicated for myself? Do I need to simplify the $f(u)$ from the course notes for a spline in a plane (eg. 2D, vs. the above which is for 3D)?
You can simplify the problem in a couple of ways:
Restrict your attention to 2D (if that's satisfactory)
Expand the matrix product, and write the curve equation in the form $$ \mathbf{X}(u) = u^3\mathbf{A} + u^2\mathbf{B} + u\mathbf{C} + \mathbf{D} $$
But I don't think you'll get a closed-form formula for the integral. You'll need to use numerical methods.
Your integrand is wrong, anyway. For parametric curves, you need to use these curvature formulae.