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}

\int_0^1 x^2 dx = \frac{1}{3} x^3 \Big |_0^1 = 1/3(1^3)- 1/3 (0)^3 = 1/3

\int_{-2}^2 (2x + 2)dx = \lim_{n \to \infty} \sum_{i = 1}^nf(x_i)\Delta x