Chapter 5.4, Problem 72E

(a)

To determine

Show that $F\left(b\right)-F\left(a\right)={\displaystyle \sum _{i=1}^n\left[F\left(x_i\right)-F\left(x_{i-1}\right)\right]}$.

(b)

To determine

Show that $F\left(b\right)-F\left(a\right)$ is a Riemann sum for $f$ on $\left[a,b\right]$ that is $F\left(b\right)-F\left(a\right)={\displaystyle \sum _{i=1}^{n}f\left(c_i\right)\Delta x_i}$.

(c)

To determine

Show that $F\left(b\right)-F\left(a\right)={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$.