This document discusses the use of integrals to calculate the areas bounded by curves. It begins by reviewing how formulas in elementary geometry can calculate the areas of simple shapes but are inadequate for figures bounded by curves. It then discusses how the definite integral, as the limit of a sum, was used in the previous chapter to find the area between a curve and the x-axis. The current chapter will look at specific applications of integrals to find the area under simple curves, as well as the area between lines and parts of circles, parabolas and ellipses. It provides an introduction to integration, defining it as the inverse process of differentiation, where an integral finds the function whose derivative is a given function.

1. Sal College of Engineering

3. In geometry, we have learnt formulae to calculate
areas of various geometrical figures including
triangles, rectangles, trapezias and circle. Such
formulae are fundamental in the application of
mathematics to many real life problems. The
formulae of elementary geometry allow us to
calculate areas of many simple figures, However,
they are inadequate for calculating the areas
enclosed by curves. For that we shall need some
concept of Integral Calculus.
In the previous chapter, we have studied to
find the area bounded by the curve y = f(x), the
ordinates x = a, x=b and x-axis, while calculating
definite integral as the limit of a sum. Here, in
this chapter, we shall study a specific application
of integrals to find the area under simple curves,
area between lines and arcs of circles , parabolas
and ellipses(standard forms only).We shall also
deal with finding the area bounded by the above
said curves.
A.L. Cauchy
(1789-1857)
INTRODUCTION

4.  Integration is the inverse process of differentiation. In the differential
calculus, we are given a function and we have to find the derivative or
differential of this function, but in the integral calculus, we are to find a
function whose differential is given. Thus, integration is a process which is
the inverse of differentiation.
Let d/ dx F(x)= f(x) . ∫f(x) dx=F (x) + C. These integrals are called
indefinite integrals or general integrals, C is called constant of
integration. All these integrals differ by a constant.
 From the geometric point of view, an indefinite integral is collection of
family of curves, each of which is obtained by translating one of the curves
parallel to itself upwards or downwards along along the y-axis.
 Some properties of indefinite integrals are as follows:
1. ∫[ f(x) + g (x)] dx =∫ f (x) dx + ∫ g (x) dx
2. For any real number k, ∫k f(x)dx = k ∫ f (x) dx
More generally, if f1,f2,f3, ….. , fn are function and k1,k2,…. , kn are real
numbers. Then
∫[k1 f1(x)+ k2f2(x)+..+kn fn(x)]dx = k1∫f1(x)dx+k2∫f2(x)dx+…+kn∫ fn(x) dx