R P Gillespie Ph.D.
Edinburgh and London
The first four chapters of this book are devoted to an elementary account of integration and demand from the student only a slight knowledge of the differential calculus. The proofs in this part of the book are based on geometrical conceptions, no attempt being made to be rigorous. In Chapter VI an account is given of infinite integrals, and in particular the properties of Gamma and Beta functions are discussed. For a student who desires a working knowledge of the integral calculus Chapters I to IV and Chapter VI cover the most important parts of the ground.
In Chapter V there is a discussion of the Riemann arithmetical definition of the integral, and Chapter VII contains further properties of the Riemann integral as well as a treatment of the Riemann double integral. For these two chapters the student is expected to have a certain amount of knowledge of elementary analysis.
I should like to record my debt of gratitude to my colleagues Miss Margaret S Black, M.A., Mr T S Graham, Ph.D., and Mr J M Hyslop, Ph.D., D.Sc., for their generous help in correcting the proofs and for their valuable suggestions. I should also like to thank the general editors of the series for their kindly advice throughout the preparation of the book.
Most of the examples are taken from examination papers set at Glasgow University.
GLASGOW, August 1939
1. The integral calculus may be said to have been begun by the Greek mathematicians who strove to evaluate the area of a circle. The area of a rectangle is equal to the product of its length and breadth, and from this, by the methods of Euclid, the areas of figures bounded by straight lines can be determined. These methods are not applicable in the case of a circle or of any figure bounded by a curve. If n-sided regular polygons are inscribed in and circumscribed about a circle, their areas can be calculated by Euclidean methods, and it is clear that the area of the circle lies between these two areas. As n increases the difference between the two areas becomes smaller, and we can make this difference as small as we please by choosing n sufficiently large. Thus we have a method which yields the area of the circle to any degree of accuracy that is required. This method is essentially that of the integral calculus. In the language of the calculus we say that the area of the circle is the limit for n tending to infinity of the area of the regular n-sided inscribed or circumscribed polygon.