Page:Calculus Made Easy.pdf/202



great secret has already been revealed that this mysterious symbol $$\int$$, which is after all only a long $$S$$, merely means “the sum of,” or “the sum of all such quantities as.” It therefore resembles that other symbol $$\Sigma$$ (the Greek Sigma), which is also a sign of summation. There is this difference, however, in the practice of mathematical men as to the use of these signs, that while $$\Sigma$$ is generally used to indicate the sum of a number of finite quantities, the integral sign $$\int$$ is generally used to indicate the summing up of a vast number of small quantities of indefinitely minute magnitude, mere elements in fact, that go to make up the total required. Thus $$\int dy=y$$, and $$\int dx=x$$.

Any one can understand how the whole of anything can be conceived of as made up of a lot of little bits; and the smaller the bits the more of them there will be. Thus, a line one inch long may be conceived as made up of $$10$$ pieces, each $$\tfrac {1}{10}$$ of an inch long; or of $$100$$ parts, each part being $$\tfrac {1}{100}$$ of an inch long;