Definite Integral

$\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(a+\frac{b-a}{n}k\right)\cdot \left(\frac{b-a}{n}\right)=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x={\int }_a^{b}f(x)\mathrm{d}x$

• The basic idea is to add a bunch of small-width rectangles with a height of $f(a+k\Delta x)$ and width of $\Delta x$
• $\frac{b-a}{n}$ is the same as $\Delta x$ seen in many textbooks
• $f(x_i^{\ast })$ the height of the rectangle as determined by some point $x_i^{\ast }$ within the width of rectangle $i$