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This book is intended for various colleges and technical universities of India. The object has been to present the Calculus and some of its important applications simply and concisely. The book will be found to be adapted to the needs of the mathematical student, and also will enable the engineer to get that knowledge of the Calculus which is required by him in order to make practical applications of the subject. All of the formulas for differentiation are established by the method of limits. This method is preferred because it is more readily understood, and rigorous than the method of infinitesimals. Moreover, it has the great advantage of being a familiar method, as the student has previously used it in Algebra and Geometry. But the differential notation is fully explained, particularly in the investigations of the Integral Calculus. Thus the essential unity of the two branches of the Calculus is emphasized. The whole subject is made more intelligible. The principal applications of the Calculus, as in Maxima and Minima, Radius of Curvature, etc., are treated at some length, while less important subjects are treated much more briefly. A large number of carefully selected examples, some original ones, and numerous practical numerical problems from mechanics and different branches of applied mathematics are given. A table of Integrals, arranged for convenience of reference, is appended. Brief Contents: Definitions and First Principles • Differentiation of Algebraic Functions • Differentiation of Transcendental Functions • Differentials and Rates • Integration • Successive Differentiation and Integration, Applications in Mechanics • Functions or Two or more Variables, Implicit Functions, Change or the Independent Variable • Development of Functions • Evaluation of Indeterminate Forms • Maxima and Minima of Functions • Tangents, Normals and Asymptotes to any plane curve • Direction of Curvature, Points of Inflection, Radius of Curvature/Contact • Evolutes and Involutes/Envelopes • Singular Points • Integration of Rational Fractions • Integration of Irrational Functions • Integration by Parts and by Successive Reduction • Integration of Transcendental Functions, Integration by Series • Integration as a Summation, Areas and Lengths of Plane Curves • Surfaces and Volumes of Solids • Centre of Mass, Moment of Inertia, Properties of Guldin • Differential Equations.