A note based on several books including Courant’s classic one “Introduction to Calculus and Analysis”.
(Theorem of Existence). For any continuous function $f(x)$ in a closed interval $[a, b]$ the integral over this interval exists as the limit of the sums $F_n$, independently of the choice of the points of the subdivision $x_1, ..., x_{n-1}$ and of the intermediate points $\xi_1, ..., \xi_{n}$ as long as the largest of the lengths $\Delta x_i$ tends to zero. The form of $F_n$ is defined as follows. For interval $[a, b]$, it is divided into $n$ sub-intervals with length $\Delta x_i = x_{i} - x_{i-1}$, where $x_0 = a$ and $x_n = b$, and $\xi_i$ represents any point in the interval $\xi_i \in [x_{i-1}, x_i]$, then $F_n = \sum_{i=1}^{n} f(\xi_i) (x_i - x_{i-1}) = \sum_{i=1}^{n} f(\xi_i) \Delta x_i$.
(Leibnitz’s Notation for the Integral). $\int_{a}^{b} f(x) dx \stackrel{\text{def}}{=} \lim_{n \to \infty} F_n = \lim_{n \to \infty} \sum_{i=1}^{n} f(\xi_i) \Delta x_i$.
Related to Lagrange’s Remainder Theorem.
(Taylor series). The Taylor series of the function $f(x)$ around $a$ in its domain is:$f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f^{(2)}(a)}{2!} (x-a)^2 + ... + \frac{f^{(n)}(x)}{n!} (x-a)^n$.
(Taylor theorem). Let $k \geq 1$ be an integer and let the function $f: \mathbb{R} \mapsto \mathbb{R}$ be $k$ times differentiable at the point $a \in \mathbb{R}$. There exists a function $h_k: \mathbb{R} \mapsto \mathbb{R}$ such that: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(x)}{k!} (x-a)^k +h_k(x)(x-a)^k$, and $\lim_{x \to a} h_k(x) = 0$. This is called the Peano form of the remainder. (based on the Wikipedia article)