Exercise 7.4.2

(a)

Let $g(x)=x^3$, and classify each of the following as positive, negative, or zero. $$\text{ (i) }{\int \; <!--\; nolimits\; -->}_0^{-1}g+{\int \; <!--\; nolimits\; -->}_0^{1}g\text{ (ii) }{\int \; <!--\; nolimits\; -->}_1^{0}g+{\int \; <!--\; nolimits\; -->}_0^{1}g\text{ (iii) }{\int \; <!--\; nolimits\; -->}_1^{-2}g+{\int \; <!--\; nolimits\; -->}_0^{1}g\text{. }$$

(b)

Show that if $b\le a\le c$ and $f$ is integrable on the interval $[b,c]$, then it is still the case that ${\int }_a^{b}f={\int \; <!--\; nolimits\; -->}_a^{c}f+{\int \; <!--\; nolimits\; -->}_c^{b}f$.