Also known as "quadrature" or "numerical quadrature" because of its origins in the ancient problem of "squaring the circle," integration is a form of continuous mathematics which was not fully possible until the advent of calculus, which deals specifically with problems just like this.
Why is it so difficult to find the area under something so simple as, say, a parabola? The nature of continuity in mathematical curves means, essentially, that there is no full and complete answer to the question of integration.
The Rectangle Rule
To understand where the challenge in integration comes from, imagine a simple problem: An upside-down parabola which crosses the x-axis at points (-2,0) and (2,0). Now, imagine being asked to find the area underneath this parabola between these two points.
In calculus classes, students learn a very simple way to write a problem like this using integral notation, but for now, it is simple enough to simply state it in words.
Without the benefit of modern mathematics, how would one go about solving a problem like this?
One simple way is to use the "rectangle rule." Begin by solving this problem for a series of points along the x-axis, perhaps for every 0.1 units. Doing this, one can essentially "divide up" the curve. Then, using these points as the midpoints for the tops of rectangles, one can essentially redraw this curve in a rising and falling series of rectangles, which looks like a simple bar graph.
The reason for dividing up the area under a curve in this way is that it is very easy to find the areas of rectangles, and so by adding the areas of all of these new polygons together, a rough estimate of the area under this curve can be obtained. The keyword here, however, is rough. Obviously, this answer cannot be considered complete, but it is a good start.
Where to go from here? The ancient mathematicians could have told you the answer: Simply make the rectangles smaller! Plot more points along the curve, and the rectangles will get smaller ans smaller, and their formation will begin to look more and more like the actual curve, thus making the sums of their areas closer and closer to the actual number!
Completing the Integral
Unfortunately, even using such a simple method as the rectangle rule to begin to find an answer to a given integral is essentially futile. As one can surely imagine, such a process could go on forever, extending to infinitely thinner and thinner rectangles, each step bringing one closer and closer to the correct answer, without every actually getting there.
Fortunately, it is here that ancient mathematics ends and modern calculus begins. The mathematical methods developed in the seventeenth century by Newton and Leibnitz finally allowed for answers to the problem of integration and got rid of these infinite progressions once and for all.
Using methods of differentiation, formal integral theory has been developed which seeks to avoid the infinities evident in problems dealing with continuous curves (similar to those problems faced when finding derivitives), and, primarily thanks to the nineteenth century mathematician Bernard Riemann, integral calculus is now taught in basic calculus classes everywhere.
The basic notion behind integral calculus, however, is not fundamentally different from the rectangle rule, except that instead of having to manually determine the area of countless tiny rectangles, calculus provides neat little tricks which essentially do an infinite amount of work with very little effort.
Isaac M. McPhee - Isaac McPhee was born as a human child in Mt. Vernon, WA, c. 1982; he currently resides in the bustling heart of New York City where he ...
