We have \[ \sum_{i=0}^n \frac{1}{2}(x_{i-1}+x_i)(x_i-x_{i-1}) = \sum_{i=0}^n \frac{1}{2}(x_i^2-x_{i-1}^2) \] and it is then clear that \[ L(P,f)\leq \sum_{i=0}^n \frac{1}{2}(x_i^2-x_{i-1}^2)\leq U(P,f) \] If you can show $U(P,f)-L(P,f)<\varepsilon$ we are done since \[ \sum_{i=0}^n\frac{1}{2}(x_i^2-x_{i-1}^2)=\frac{1}{2}(b^2-a^2) \]