Fundamental theorem of calculus

The fundamental theorem of calculus tells us that, given some function $f$ of a given regularity class, then, if there exists a function $F$ such that $F' = f$, we have

\begin{equation} \int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a) \end{equation}

In the Riemann integral

In the case of the Riemann integral, we consider a partition of the integration interval $[a,b]$, that is, we consider some set $\{ x_i \}$, $0 \leq i \leq N$, with $x_0 = a$, $x_N = b$, and for every value of $i$,

\begin{equation} x_i < x_{i+1} \end{equation}

The lower and upper Riemann sum of that partition $P$ are defined as

\begin{eqnarray} L_f &=& \sum_{i = 1}^N (x_i - x_{i-1}) \min_{x \in [x_{i-1}, x_i]} f(x)\\ U_f &=& \sum_{i = 1}^N (x_i - x_{i-1}) \min_{x \in [x_{i-1}, x_i]} f(x) \end{eqnarray}