Page:Calculus Made Easy.pdf/203

or of $1,000,000$ parts, each of which is ${\tfrac {1}{1,000,000}}$ of an inch long; or, pushing the thought to the limits of conceivability, it may be regarded as made up of an infinite number of elements each of which is infinitesimally small.

Yes, you will say, but what is the use of thinking of anything that way? Why not think of it straight off, as a whole? The simple reason is that there are a vast number of cases in which one cannot calculate the bigness of the thing as a whole without reckoning up the sum of a lot of small parts. The process of “integrating” is to enable us to calculate totals that otherwise we should be unable to estimate directly.

Let us first take one or two simple cases to familiarize ourselves with this notion of summing up a lot of separate parts.

Consider the series:

$1+{\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+{\tfrac {1}{16}}+{\tfrac {1}{32}}+{\tfrac {1}{64}}+\;$ etc.

Here each member of the series is formed by taking it half the value of the preceding. What is the value of the total if we could go on to an infinite number of terms? Every schoolboy knows that the answer is $2$. Think of it, if you like, as a line. Begin with

Fig. 46.

one inch; add a half inch, add a quarter; add an eighth; and so on. If at any point of the operation