Are Calculus And Numerical Analysis Obsolete?

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Are Calculus And Numerical Analysis Obsolete?

By Thomas Keyes
May 22, 2005

Anyone who has studied conventional calculus knows that the two procedures call differentiation and integration are mutually inverse. If we integrate an _expression and differentiate the outcome, or viceversa, we should be back to what we began with. An easy case is a so-called polynomial, as, for example, the general _expression for a parabola, which is:

(1) Y = A*X^2 + B*X + C

Integrating this, we get the following indefinite integral:

(2) int (Y*dY) = 1/3*A*X^3 + ½*B*X^2 + C*X + D

If we differentiate this, we get our original _expression, (1). Likewise, if we differentiate (1), we get the following derivative:

(3) dY / dX = 2*A*X + B

And if we integrate this, we again get our original _expression, (1).

A parabola is a curve of indefinite length. We can extend it to plus or minus infinity. The integral gives us the area under the parabola, and we usually want to know the area between two points along the parabola. So we specify limits, as for example, we specify the area under the curve, from the point where X = 4 to the point where X = 10. Then in equation (2), we substitute for X, first 10, and then 4, subtracting the latter result from the former, to get our answer. An integral with specified limits is called a definite integral.

One who has studied calculus knows also that differentiation is relatively easy and can be accomplished by following definite procedures. But integration is often difficult. Actually, in most cases there are no procedures other than trial and error. You must guess the integral and then differentiate to see if your guess is right. That is why one often must consult, usually unavailingly, a table of integrals, if he finds an _expression that he cannot readily integrate himself. Over the centuries, mathematicians have found hundreds of integrals by happenstance that cannot be worked out methodically. And these have been collected and published.

However, in many cases, no integral exists at all. Not knowing this, you could spend hours and days trying unsuccessfully to integrate an _expression.

What if you need to know the answer? Suppose you have a curve that is going to form part of your calculations for a piece of machinery or a structure. If you cannot integrate for the length of the curve, you are stymied. So you resort to a procedure called numerical analysis.

Basically, in this case, numerical analysis amounts to setting up a series of right triangles whose hypotenuses approxímate the curve whose length you are seeking. You divide the curve into ten little triangles, calculate ten hypotenuses and add up the lengths. How good is your answer? To find out, you try 20 little triangles. You get a somewhat different answer. So you try 30. You get yet another answer, which varies less from the second answer than the second varied from the first. So you try 40, and the variation is still less. So you try 50, 60, 70, etc. Eventually the variation begins to vanish, at least for the number of decimal places that you’re working with. So you conclude that you now have an adequately precise answer. This vanishing of the difference is called convergence.

Of course, this is all very time-consuming, especially with pencil and paper and a book of logarithms, but even with a calculator that has only basic arithmetical operations.

But with a good caluclator, like TI-80 or TI-85, you can calculate definite integrals instantaneously. You input the _expression to be integrated, like (1) above, and you input the limits of integration, as, for example, 10 and 4, as mentioned above. Of course, there is a specified way to input these parameters, which must be learned from the instruction manual that accompanies the calculator.

The calculator does not solve for the indefinite integral. In other words, a formula corresponding to (2) above does not appear in the display window. Rather, the numerical value of the definite integral is displayed. You get your numerical answer, but you don’t get the formula for the indefinite integral, even if this is known to exist. And I assume that the method used by the calculator is very much like the method of numerical analysis, performed at the speed of light in a microscopic circuit.

Using such a calculator I will never know the general formula for the area under a parabola, but I’ll be able to get the numerical value for any given parabola, accurate to more decimal places than are likely ever to be needed in computations relative to ordinary machinery, electrical circuits, structures, acoustics and the like.

So why do I need calculus at all?

This is reminiscent of the old problems of squaring the circle and trisecting an angle, neither of which can be done using “classical” methods. But if I can get an answer accurate to 10 or 12 decimal places, what do I care about “classical” methods?

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About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far.

I have studied languages for years and traveled extensively on five continents.

Email: udikeyes@yahoo.com

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