Areas using Integrals

                                     Bounded Areas using Integration
The Integral  represents area bounded by the curve y=f(x) on one side, the line x=a and x=b on other two sides, x-axis on the fourth side. 

If the area lies below x-axis gives negative value as area. Therefore the formula in such case is,   Area=   or   

The Integral   represents area bounded by the curve x=f(y) on one side, the line y=c and y=d on other two sides, y-axis on the fourth side.          

If the area lies left of y-axis   gives negative value as area. Therefore the formula in such

case is,   Area=   or  

When using Integrals to find the Area we always get the area between the curve and the axis (dx or dy). For example, (refer the figure below) while finding area bounded by a parabola open upwards from x=a to x=b and x-axis we will not get area A1 but we arrive at area A2.

If the Area under consideration lies above as well as below x-axis (see figure below), then you have to find area like this,  or


  

                                                              

A Set is  a collection of objects.
In maths it is a collection of numbers.

A Cartesian Product is combination of 2 or more sets. it is represented as AxB
for example A={a,b,c} B={1,2,3}
then AxB={(a,1) (a,2) (a,3) (b,1) (b,2) (b,3) (c,1) (c,2) (c,3)}

A Relation is  a subset of AxB.
The first set is called Domain.
Second Set is called Co-domain.
In other words a Relation is a random combination of elements of Domain to Co-domain.

A Function is special relation which should follow 2 rules.
1. No element should be leftout in domain without mapping. ie., all elements in the
    domain should be in relation.
2. one element in domain cannot be mapped to two or more elements in
     co-domain. (note. But two or more elements in the domain can be mapped
     to one element in the co-domain.)
 

Note: All Functions are Relations but all Relations are not Functions.