Chapter 5.2, Problem 130E

Suppose [a, b] can be subdivided into subintervals $a=a_0<a_1<a_2<\mathrm{...}<a_N=b$ such that either $f\ge 0$ over $\left[a_{i-1},a_i\right]$ or $f\le 0$ over $\left[a_{i-1},a_i\right]$ . Set $A_i={\displaystyle {\int }_{a_{i-1}}^{a_i}f\left(t\right)dt}$ .

a. Explain why ${\displaystyle {\int }_a^{b}f\left(t\right)dt=A_1+A_2+\mathrm{...}+A_N}$ .

b. Then, explain why $\left|{\displaystyle {\int }_a^{b}f\left(t\right)dt}\right|\le {\displaystyle {\int }_a^b\left|f\left(t\right)\right|dt}$ .