riemann sums

$A\approx L_n=f(x_0)\Delta x+f(x_1)\Delta x+\cdots +f(x_n-1)\Delta x={\sum }_{i=1}^{n}f(x_{i-1})\Delta x$

${\int }_a^{b}f(x)dx$

${\lim }_{n\to \infty }{\sum }_{i=1}^{n}f(a+\frac{(i-1)(b-a)}{n})\cdot \frac{(b-a)}{n}$
height: $f(a+\frac{(i-1)(b-a)}{n})$
width: $\frac{(b-a)}{n}$

$R_n$ is $L_n$ but ending at $i$ instead of $i-1$

${\int }_a^b[f(x)+g(x)]={\int }_a^{b}f(x)+{\int }_a^{b}g(x)$
${\int }_a^{b}cf(x)=c{\int }_a^{b}f(x)$
${\int }_a^{b}f(x)={\int }_a^{c}f(x)+{\int }_c^{b}f(x)$

average value: $f_{avg}=\frac{1}{b-a}{\int }_a^{b}f(x)$