integral

Def 29.3

$${\int }_a^{b}f(x)dx=I=\underset{k}{\lim }{\overline{\sum }}_{{\tau }_k}f=\lim R(f,\xi )$$

if $f$ is integrable

Prop 29.4

$\forall \sigma$

1. ${\underline{\sum }}_{\sigma }f\le {\int }_a^{b}f(x)dx\le {\overline{\sum }}_{\sigma }f$
2. $|{\int }_a^{b}f(x)dx|\le {\overline{\sum }}_{\sigma }|f|$
3. $|{\int }_a^{b}f(x)dx|\le {\int }_a^b|f(x)|dx$

Thm 30.3

for $f$, $g$ is integrable

1. ${\int }_a^b(f(t)+\lambda g(t))dt={\int }_a^{b}f(t)dt+\lambda {\int }_a^{b}g(t)dt$
2. ${\int }_a^{b}f(t)dt={\int }_a^{c}f(t)dt+{\int }_c^{b}f(t)dt$

Thm 30.5

$$|{\int }_a^{b}fdt|\le {\int }_a^b|f|dt$$

proved by considering $\Phi (y)=|y|$ as a Lipschitz function.

Def 31.5

By Oresme

$${\int }_0^1{x}^{m}dx=\frac{1}{m+1}$$

Prop 41.3

$${\int }_a^{b}f(s)ds\le (b-a)\underset{s}{\sup }f(s)$$