Mathematics · Ch 17 — Definite Integrals
Introduction
Introduction
The Motivation for Integral Calculus
Differential calculus centres on the derivative, originally motivated by the problem of defining tangent lines to graphs and calculating their slope. Integral calculus, by contrast, is motivated by the problem of defining and calculating the area of the region bounded by the graph of a function.
Consider a function that is differentiable on an interval — that is, its derivative exists at every point of . A natural question then arises: given at each point of , can we determine the original function ? The functions that could have a given function as their derivative are called anti-derivatives (or primitives) of that function. The formula giving all these anti-derivatives is the indefinite integral, and the process of finding them is called integration.
The term "primitive" is historical — these functions are the "original" or "first" functions from which the derivative came.
This type of problem arises in many practical and theoretical situations. For instance, if we know the instantaneous velocity of an object at any instant, can we determine its position at any instant?
The Two Fundamental Problems
Integral calculus arises from efforts to solve two types of problems:
- Finding a function whenever its derivative is given — this leads to the indefinite integral.
- Finding the area bounded by the graph of a function under certain conditions — this leads to the definite integral.
Together, these two forms — the indefinite integral and the definite integral — constitute what we call Integral Calculus.
The Fundamental Theorem of Calculus
There is a connection, known as the Fundamental Theorem of Calculus, between the indefinite and definite integral. This theorem makes the definite integral a practical tool for science and engineering, and it is also used to solve many problems in economics, finance, and probability.
The Fundamental Theorem of Calculus is the bridge connecting the two branches of calculus — it shows that differentiation and integration are inverse operations.
Scope of This Chapter
In this chapter, we confine ourselves to indefinite and definite integrals, their elementary properties, and some techniques of integration. The formal definition of the indefinite integral, its notation, and the constant of integration are developed next.