Definition of the integral
One can show that, as long as the function \(f(x)\) is continuous (see below), then using any of these estimates, as we take more and more thinner and thinner rectangles (or trapezia) — i.e., as \(n \to \infty\) or \(\Delta x \to 0\) in the limit — the area estimates computed will converge to an answer for the exact area under the graph. Moreover, no matter which estimate we use, we will end up with the same answer.
For our purposes, continuous means that you can draw the graph \(y=f(x)\) without taking your pen off the paper. Actually, it's sufficient that the graph is piecewise continuous, i.e., constructed out of several continuous functions.
For a function \(f(x)\), considered between \(x=a\) and \(x=b\), the limit we obtain for the exact area is denoted by\[ \int_a^b f(x) \; dx \]
and is called the definite integral, or simply the integral, of \(f(x)\) with respect to \(x\) from \(x=a\) to \(x=b\). The symbol \(\int\) is called an integral sign, and the numbers \(a,b\) are called the terminals or endpoints of the integral. The function \(f(x)\) is called the integrand.
We can think of the integral as being the limit of a sum. Roughly speaking, the area estimates are given by expressions like \(\sum f(x_j) \; \Delta x\) and in the limit these become \(\int f(x) \; dx\)