Differential and Integral Calculus

Abstract

Differential and Integral Calculus are two fundamental branches of mathematical analysis that focus on understanding the behaviour of functions and their applications to real-world problems. Differential calculus concerns the study of rates of change and the concept of the derivative, while integral calculus deals with the accumulation of quantities and the concept of the integral. Together, they provide powerful tools for modelling and solving problems in physics, engineering, economics, biology, and other fields. Integral calculus, on the other hand, focuses on the concept of the integral as a method for finding areas under curves, volumes of solids, and accumulated quantities. The Fundamental Theorem of Calculus, which links differentiation and integration, is emphasized as a pivotal result. Methods for solving definite and indefinite integrals, including integration by parts, substitution, and numerical methods, are examined alongside their applications in areas such as physics (e.g., work, energy), probability theory, and economics. The Book also highlights the importance of multivariable calculus, which extends these concepts to functions of several variables, and discusses the role of differential and integral calculus in the development of mathematical models that describe real-world phenomena.