Integrating a Piecewise Function

It is necessary to split the integral over the intervals [0,1], [1,2], and [2,3] where the function takes different forms.

The total area under $f(x)$ on [0,3] can be computed using a single integral $\int_{0}^{3} f(x)\,dx$ without separating the intervals.

After computing the integrals over each interval, their sum gives the total area under the curve.

A piecewise function requires evaluation of separate integrals over each section where the functional form changes.