# Formal Definition
Let $f$ be a continuous function on an interval $[a, b]$.

Divide $[a, b]$ into $n$ equal parts of width $\Delta x = \dfrac{b-a}{n}$.

Let $x_0, x_1, x_2, \cdots, x_n$ be the endpoints of this subdivision. $x_0 = a$ and $x_n = b$. 

Define $$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^nf(x_i)\Delta x$$
- $\Delta x$ refers to the width of each sub-interval, or $\frac{b-a}{n}$.
- $f(x_i)$ refers to the height of each subinterval, and can be found with the equation $x_i = \Delta xi + a$ 
	- Or, the width of each interval times the interval index, plus the starting offset.

Then let $f$ be a continous function on $[a, b]$ and let $F$ be the antiderivative of $f$ (i.e $F'(x) = f(x)$).

Then $\int_a^b f(x) dx = F(b) - F(a)$.

## Sum of an infinite series
The sum of an infinite series can be defined as follows:
$$ \sum_{i = 1}^n i = \frac{n(n+1)}{2}$$

## Examples
1. Find the area under the curve between 0 and 1 of the function $f(x) = x^2$
$$ \int_0^1 x^2 dx = \frac{1}{3} x^3 \Big |_0^1 = 1/3(1^3)- 1/3 (0)^3 = 1/3$$
2. Find the Riemann Sum under the curve between -2 and 2 of the function $2x + 2$. 
$$ \int_{-2}^2 (2x + 2)dx = \lim_{n \to \infty} \sum_{i = 1}^nf(x_i)\Delta x $$
