Definite Integral

#Math

$\displaystyle \lim_{ n \to \infty }\sum_{k = 1}^{n}f\left( a+\frac{b-a}{n}k \right)\cdot\left( \frac{b-a}{n} \right)=\lim_{ n \to \infty }\sum_{i = 1}^{n}f(x_{i}^{*})\Delta x=\int_{a}^{b} f(x) , \mathrm{d}x$

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