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.