#### Chapter 8 Integral Calculus

Section 8.1 Antiderivatives

# 8.1.1 Introduction

In the previous module we studied derivatives of functions. As for every other arithmetic operation,the question of finding an inverse operation arises. For example, subtraction is the inverse operation of addition, and division is the inverse operation of multiplication. The question of the inverse operation of differentiation leads to the introduction of integral calculus and the definition of an antiderivative. The relation between derivative and antiderivative can be explained very easily. If a derivative $f\text{'}$ can be assigned to a function $f$, and the derivative $f\text{'}$ is also considered as a function, then the function $f$ could be assigned to this function $f\text{'}$ by inverting the operation of "differentiation". Thus, in this chapter the question is: for a given function $f$, can we find another function with $f$ as its derivative?
The applications of integral calculus are as varied as the applications of differential calculus. In physics, for example, the force $F$ acting on a mass $m$ may be investigated. From the well-known relation $F=ma$ ($a$ being the acceleration of the object), the acceleration $a=F/m$ can be calculated from the force. If the acceleration is interpreted as the rate of velocity change, i.e. $a=\frac{dv}{dt}$, then the velocity can be determined subsequently from the inverse of the derivative - from integral calculus. Similar relations can be found in many fields of science and engineering, and also in economics. Integral calculus is used for the calculation of areas, centres of mass, bending properties of beams or the solutions of so-called differential equations, which are used so frequently in science and engineering.