The integral $$\begin{equation} I=\int_a^bf(x) dx \tag{1} \end{equation}$$ has a very simple meaning. The integral is the area enscribed by the function $f(x)$ starting from $x=a$ to $x=b$. It is subdivided in several smaller areas whose evaluation is to be approximated by different techniques. The areas under the curve can for example be approximated by rectangular boxes or trapezoids.