When f is a non-negative function and a < b, the area between the graph of f and the x-axis on [a, b] is called the integral of f from a to b with respect to x. It's written like this:
The funny symbol S-shaped symbol is called an integral sign. The function f(x) is called the integrand. The dx part means that x is the variable of integration. The integral is taken from the number at the bottom of the integral sign
to the number at the top of the integral sign
.
The number at the bottom of the integral sign is called the lower limit of integration, while the number at the top of the integral sign is called the upper limit of integration.
When the area between the function and the horizontal axis isn't made up of nice shapes, it's harder to find the integral. That's why we spent all that time approximating areas: the left-hand sums and right-hand sums and midpoint sums and trapezoid sums are ways to approximate the value of a definite integral.
Let f be a non-negative function on [a, b]. As the number of sub-intervals increases, the left-hand sum approaches the actual area between f and the x-axis on [a, b]. Translating into symbols, as n approaches ∞, LHS(n) approaches 
. We defined as the area between the graph of f and the x axis on [a, b], but we can use this discussion to get a more precise definition.
Define the integral of f on [a, b] as
.
The same discussion holds for the other kinds of sums, so we could also define the integral of f on [a, b] using any of the following statements:
If you like, you can think of that dx in the integral notation as a Δ x that has gotten even smaller. If we could have infinitely many sub-intervals, each of them would have width dx.