Chapter 5.6, Problem 35ES

A function $f\left(x\right)$ is defined piecewise on an interval. In these exercises: (a) Use Theorem $\mathrm{5.5.5}$ to find the integral of $f\left(x\right)$ over the interval, (b) Find an antiderivative of $f\left(x\right)$ on the interval, (c) Use parts (a) and (b) to verify Part $1$ of the Fundamental Theorem of Calculus.

$f\left(x\right)=\left\{\begin{array}{cc}x,& 0\le x\le 1\\ x^2,& 1<x\le 2\end{array}\right.$