Riemann Integral

Definition

Riemann Sum

$$\sum _{i=1}^{n-1}f(t_i)(x_{i+1}-x_i)$$

Riemann Integral

$\forall ϵ,\exists \delta >0,\forall x_0,\dots ,x_n$ and $\forall t_1,\dots ,t_{n-1}$, as long as $\lambda =\max (x_{i+1}-x_i)<\delta$, then $|{\sum }_{i=1}^{n-1}f(t_i)(x_{i+1}-x_i)-S|<ϵ$

Then $S$ is the limit of Riemann sum

Properties

$${\int }_a^b(\alpha f(x)+\beta g(x))dx=\alpha {\int }_a^{b}f(x)dx+\beta {\int }_a^{b}g(x)dx$$

This is due to the definition of Riemann integral and linear properties of limits. Rules with Real Number Operations