Previous examples have been examples of indefinite integrals or anti-derivatives - evaluations that represent an infinite number of solutions distinguished by the constant of integration 'C'. A definite integral is an anti-derivative on a function over known bounds, and evaluates to a real number.
The first step in considering definite integrals is to look at the area of a function above the 'x' axis.
If the function is continuous over a closed interval [a,b], and the interval is divided into a number of divisions of equal horizontal lenght (Δx), we can calculate a Riemann sum which is the sum of all the products of the interval length, and the value of the function at each value of 'x' (f(x)).
This gives us an estimate of the area under the curve f(x), and may be positive, negative, or 0, depending on the behavior of the curve relative to the 'x' axis. Algebraically, the Riemann sum of f(x) for 'n' equal intervals of 'x' over the interval [a,b],is expressed as:
${S}_{n}=f\left({x}_{1}\right)\Delta x+f\left({x}_{2}\right)\Delta x+f\left({x}_{3}\right)\Delta x+\text{....}+f\left({x}_{n}\right)\Delta x$
$\text{Or}\phantom{\rule{15px}{0ex}}{S}_{n}=\sum _{i=1}^{n}f\left({x}_{i}\right)\Delta x$
This exercise illustrates how the Riemann sum can be evaluated using four equal sub-intervals, [0,2], where xi is the right endpoint of the interval, on the function: $f\left(x\right)={x}^{2}$
Riemann Sum Example