A definite integral of a function $f(x)$ over the interval $(0,1)$ can be (somewhat simplistically) written as follows: $$ \int_0^1 f(x)\,dx=\lim_{\Delta x\,\to\, 0^+} \left(\,\sum _{x=0,\,\Delta x,\,2\Delta x,\,\dots}^1 \Delta x\, f(x)\right) $$ This equation shows that the integral is the limit of increasingly fine-grained discrete sums approximating it.
I am working on a physical problem where the solution can be found using a very similar process: given a real-valued piecewise smooth function $f(x)$ defined on the interval $(0, 1)$, we divide the interval into small fragments of length $\Delta x$, compute a square matrix of a special form for each fragment, take the matrix product of all the matrices, and finally, take the limit of that product for $\Delta x\to0^+$. The notation is borrowed from here; as usual, we write $dx$ to hint at an infinitesimal $\Delta x$:
$$ \prod_0^1\exp\!\left(\left[ \begin{array}{cc} 0 & -1 \\ f(x) & 0\\ \end{array} \right]\right)^{\!dx}=\lim_{\Delta x\,\to\, 0^+} \left(\,\prod _{x=0,\,\Delta x,\,2\Delta x,\,\dots}^1 \exp\left(\Delta x\left[ \begin{array}{cc} 0 & -1 \\ f(x) & 0\\ \end{array} \right]\right)\right)$$ The shorter the fragments are, the more matrices are in the product, the closer each matrix is to the identity matrix, and the closer their product is to the exact solution. I wonder if there is a name for this mathematical concept and any theory around it, including methods for symbolic and numeric computations of such limits. Because the matrices in the product do not commute, I do not see an obvious way to reduce it to an integral.
Note: The matrix exponent from the product can be evaluated as follows: $$\small\exp\left(\left[ \begin{array}{cc} 0 & -1 \\ f(x) & 0\\ \end{array} \right]\right)^{\!h}=\exp\left(h\left[ \begin{array}{cc} 0 & -1 \\ f(x) & 0\\ \end{array} \right]\right)=\left[ \begin{array}{cc} \cos \left(h\sqrt{f(x)}\right) & -h\operatorname{sinc} \left(h\sqrt{f(x)}\right) \\ \sqrt{f(x)} \sin \left(h\sqrt{f(x)}\right) & \cos \left(h \sqrt{f(x)}\right) \\ \end{array} \right]$$