Chapter 5.5, Problem 41ES

Let $f\left(x\right)=C$ be a constant function.

(a) Use a formula from geometry to show that

${\displaystyle {\int }_a^{b}f\left(x\right)dx=C\left(b-a\right)}$

(b) Show that any Riemann sum for $f\left(x\right)$ over $\left[a,b\right]$ evaluates to $C\left(b-a\right)$ . Use Definition $\mathrm{5.5.1}$ to show that

${\displaystyle {\int }_a^{b}f\left(x\right)dx}=C\left(b-a\right)$