Question

What is the midpoint rule formula?

Answer 1

Midpoint Rule:

The function $f\left(x\right)$ is continuous on the interval $\left[a,b\right]$, then the midpoint formula for approximating the definite integral is,

${\int }_a^{b}f\left(x\right)dx\approx \sum _{i=1}^{n}f\left(\overline{x_i}\right)△x=△x\left[f(\overline{x_1)}+f(\overline{x_2)}+f\left(\overline{x_3}\right)+....+f\overline{\left(x_n\right)}\right]$

Where $△x=\frac{b-a}{n}$

And $\mathrm{midpoint}\mathrm{of}\left(x_{i-1},x_i\right)=\overline{x_i}=\frac{1}{2}\left(x_{i-1}+x_i\right)$