With minor edit with respect to the comments.
Suppose I want to evaluate \begin{align} \int^{T}_0{e^{-t}x(t)dt}. \end{align} However, I can only approximate the continuous function $\mathbb{R}_+ \ni t \mapsto x(t)$ by a discrete sequence $\{x_t : t \in \{0,\Delta t, 2\Delta t, \ldots,N\}\}$ where $\Delta t := T/N$.
- Is the following true (given $N \to \infty$)? \begin{align} \int^T_0{e^{-t}x(t)dt} \approx \sum^{N}_{t=0}{e^{-t}x_t\Delta dt} \end{align}