Suppose you want to determine the area A under the curve y = f(x) between x = a and x = b. An analytic solution to this problem would find an antiderivative F(x) for f(x). Then, using the Fundamental Theorem of Calculus, you find the desired area by evaluating the exact mathematical expression for the solution given by
A = F(b) - F(a).Provided that f(x) is simple enough for you to calculate the antiderivative, you can use this method. In many real-world problems, however, f(x) is too complex to antidifferentiate. You then apply a numerical method. In the example above, you may approximate the area by the Trapezoidal Rule, for example:
A* = h*{.5*y(0) + y(1) + y(2) + ... + y(n-1) + .5*y(n)).Instead of calculating A, for which we have no expression that we may numerically evaluate, we calculate A*, an approximation to A. (In the above example, I divided the interval [a,b] into n sub-intervals, each of width h, and drew a trapezoid lying under y = f(x) in each sub-interval. y(i) is f(a+i*h), i=0,...,n, the height of the left-most altitude of the ith trapezoid. A* is the sum of the areas of the trapezoids which lie under the curve y = f(x). See An Introduction to Numerical Analysis [2] or any introductory numerical analysis text.)In both cases, the goal is the same — evaluate the integral of f(x) over [a,b]. The methods, however, are different. In the first case, you have an exact mathematical formulation for the solution. In the second case, you must live with a formulation that is only an approximation to our goal.
In both case, of course, the final answer is subject to the numerical error that finite-precision calculating machines introduce.