Riemann Sums

What?

It’s the formula for calculating the sum of all of those rectangles under a function (Leading to the idea of Integrals).

Riemann Sums and Area Under a Curve

• Let $f(x)$ be a continuous function and $P$ is a subsection in the interval
• Let $\Delta x$ be the width of each subinterval $[x_{i-1},x_i]$ → $\Delta x=\frac{b-a}{n}$
• For each $i$, let $x_i^{\ast }$ be any point in $[x_{i-1},x_i]$. → $x_i=a+\Delta x\cdot i$

A Riemann sum is defined for $f(x)$ as

$$\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$$

The area under the curve where $y=f(x)$ is a continuous, non-negative function on an interval $[a,b]$ is given by:

$$A=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$$

(As the sum grows, the delta shrinks, the approximation becomes more accurate)