So I was reading this paper by a chemist where he used trapezoid approximation to find value of integration of an unknown function $\ f(t) = \int [A]^\alpha dt$. For context:
The method presented here uses the popular graphical interrogation of kinetic data, which takes advantage of the human visual capacity to easily, quickly, and accurately identify tendencies and patterns. The profiles of experiments differing in the concentration of one reactant, A, will only overlay when the time axis is replaced by the time integral of the concentration of A raised to the correct power α [Eq. 1]. This function is a priori unknown, but it can be approximated by using the trapezoid rule [Eq. (1)]. All of the values necessary to apply this formula are known, either because they have been measured experimentally or because they can be deduced from the concentration profiles of another reaction component if the stoichiometry is known. Therefore, the construction of the new time scale is easy and quick to perform by using any spreadsheet, without the need for sophisticated kinetic analysis packages.
He then presented the following equation:
$$\int_{t=0}^{t=n} [A]^\alpha dt = \sum_{i=1}^n(\frac{[A]_i+[A]_{i-1}}{2})^\alpha(t_i-t_{i-1})$$
$[A]_i$ is concentration of A at any timepoint, which can be determined experimentally during reaction progress and $\alpha$ is any arbitrary number that corresponds to reaction order.
The equation seems very odd to me, doesn't it supposed to be $$\int_{t=0}^{t=n} [A]^\alpha dt = \sum_{i=1}^n(\frac{[A]_i^\alpha+[A]_{i-1}^\alpha}{2})(t_i-t_{i-1})$$
Or am I missing something? His subsequent paper used the same equation and applied the exponent after $\frac{[A]_i+[A]_{i-1}}{2}$.
Assuming $[A]=[A](t)$ is continuous, then for some $t^*_i\in[t_{i-1},t_i]$, $[A](t_i^*)$ will have value $\frac{[A]_{i-1}+[A]_i}{2}$. Then it is just the Riemann sum $$ S(f;P)=\sum_i f(t_i^*)(t_i-t_{i-1}) $$ for $P$ this marked partition of $[t_0,t_n]$.