In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo integration.

Imagine you have a curve on a graph, and the curve stays above the x-axis between two points, a and b. The area under that curve, from a to b, is what we want to figure out. This area can be described as the set of all points (x, y) on the graph that follow these rules: a ≤ x ≤ b (the x-coordinate is between a and b) and 0 < y < f(x) (the y-coordinate is between 0 and the height of the curve f(x)). Mathematically, this region can be expressed in set-builder notation as $${\displaystyle S=\left\{(x,y)\,:\,a\leq x\leq b\,,\,0<y<f(x)\right\}.}$$

To measure this area, we use a Riemann integral, which is written as: $${\displaystyle \int _{a}^{b}f(x)\,dx.}$$

This notation means “the integral of f(x) from a to b,” and it represents the exact area under the curve f(x) and above the x-axis, between x = a and x = b.

The idea behind the Riemann integral is to break the area into small, simple shapes (like rectangles), add up their areas, and then make the rectangles smaller and smaller to get a better estimate. In the end, when the rectangles are infinitely small, the sum gives the exact area, which is what the integral represents.

If the curve dips below the x-axis, the integral gives a signed area. This means the integral adds the part above the x-axis as positive and subtracts the part below the x-axis as negative. So, the result of ${\displaystyle \int _{a}^{b}f(x)\,dx}$ can be positive, negative, or zero, depending on how much of the curve is above or below the x-axis.

A partition of an interval [a, b] is a finite sequence of numbers of the form $${\displaystyle a=x_{0}<x_{1}<x_{2}<\dots <x_{i}<\dots <x_{n}=b}$$

Each [x_i, x_{i + 1}] is called a sub-interval of the partition. The mesh or norm of a partition is defined to be the length of the longest sub-interval, that is, $${\displaystyle \max \left(x_{i+1}-x_{i}\right),\quad i\in [0,n-1].}$$