e to an integral as an infinite product via the definition of the integral

Question

Is there a name for this relationship? I am having a hard time searching for it. I'm hoping I've typed this up correctly, it's my first question here.

$x_k^\star$ is for instance $\frac{k \cdot h}{n}$

$$\lim \limits_{n \to \infty}\left[\prod \limits_{k=0}^n\left(1+f(x_k^\star)\cdot\frac{h}{n}\right)\right] = e^{\int \limits_{0}^hf(x) \, dx}$$

Answer 1

Because of the way your $x_k^\star$ behave (they actually depend on $n$ as well as $k$), this will usually not be an "infinite product".

An interesting related subject studies "product integrals" like the left side of your equation.

Answer 2

Essentially, use that $$ \ln\prod_{k=0}^n(1+ a_k)=\sum_{k=0}^n\ln(1+a_k)$$ together with $\ln(1+ a_k)\approx a_k$ when $a_k\approx 0$. (One has to make the latter a bit more explicit though)