Concepts, Problems and Solutions in Integral Calculus : 2000+ Solved Problems.

In writing the present treatise on the Integral Calculus the object has been to produce a work at once elementary and complete adapted for the use of beginners, and sufficient for the wants of advanced students. In the selection of the propositions, and in the mode of establishing them, we have endeavored to exhibit fully and clearly the principles of the subject, and to illustrate all their most important results. The process of summation has been repeatedly brought forward, with the view of securing the attention of the student to the notions which form the true foundation of the Integral Calculus itself, as well as of its most valuable applications. Considerable space has been devoted to the investigations of the lengths and areas of curves and of the volumes of solids, and an attempt has been made to explain those difficulties which usually perplex beginners especially with reference to the limits of integrations.

The transformation of multiple integrals is one of the most interesting parts of the Integral Calculus, and the experience of teachers shows that the usual modes of treating it are not free from obscurity. We have therefore adopted a method different from those of previous elementary writers, and have endeavored to render it easily intelligible by full details, and by the detailed solutions of a large number of problems, which will serve as a complete guide to private students reading the subject with few or no opportunities of instruction.

The Calculus of Variations seems to claim a place in the present treatise with the same propriety as the ordinary theory of maxima and minima values is included in the Differential Calculus. Accordingly a chapter of the treatise is devoted to this subject ; and it is hoped that the theory and illustrations there given will be found, with respect to simplicity and comprehensiveness, adapted to the wants of students.

In order that the student may find in the volume all that he requires, a large collection of problems for exercise has been appended to the several chapters. These examples have been selected from the College and University Examination Papers, and have been verified, so that it is believed that few errors will be found among them.

The work has been carefully revised since its first appearance, and additions made to it with the hope of increasing its utility for the purposes of instruction, and of rendering it still more worthy of the favor with which it has been received.

Solutions are provided next to the problems throughout the book. This will save the time and lighten the work of Teachers as well. This book helps in acquiring a better understanding of the basic principles of integral calculus and in revising a large amount of the subject matter quickly. Care has been taken, as in the former Keys, to present the solutions in a simple natural manner, in order to meet the difficulties which are most likely to arise and to render the work intelligible and instructive.

The problems have been selected with a view to illustrate every part of the subject, and, as the number of them is more than two thousand, we trust they will supply ample exercise for the student. Complicated and difficult problems have been excluded, because they consume time and energy which may be spent more profitably on other branches of mathematics. Each set of examples has been carefully arranged, commencing with some which are very simple and proceeding gradually to others which are less obvious. The detailed solutions to all the problems are provided.

This work contains several variations of problems, solutions, methods, approaches to enrich, strengthen and enliven the inherent multi-concepts.