Definition of the Integrals

What is an Integral?

Example:

An acquisition officer has observed the number of people allocated to a project over a period of 30 months. The following table depicts the values obtained over that period.

Time(in Months)	No. of People
5	320
10	630
15	1120
20	1600
25	1810
30	1920

Using these values, he plotted the graph of NUMBER OF PEOPLE with respect to TIME (in MONTHS).

He considered the NUMBER OF PEOPLE as N(t) and TIME (IN MONTHS) as t. Now the acquisition officer is interested in finding the number of persons allocated during the project over the period of 30 months.

This problem can be solved using the concept of integrals. As we can see from the graph above, the area beneath the curve N(t) is in an irregular shape. But, we can approximate it by imposing rectangles on it as shown in the graph below. Now, using the concept of integrals, we divide the required area into many smaller rectangles with very small width

This

x can be centered at any point "c" and height "N(c)". Now if we take the integral of the curve N(t) from t=0 to t=30, we get the required area.

Definition:

In many problems the derivative or differential of a function is known and it is necessary to find the function. For instance, consider the speed of a moving particle at time t is

ds/dt = f(t)
And the distance it moves in time 't' is a quantity 's' with differential
ds = f(t) dt.
To find the distance we must then find a function of 't' with differential equal to f(t) dt.
The process of finding a function with given differential is called integration. If
d F(x) = f(x) dx
then F(x) is called an integral of f(x)dx and this is indicated by the notation
F(x) = f(x) dx.

From the above statements we can say that, by differentiation we pass from a function to its differential. By integration we pass from the differential to the function. Integration is thus the inverse of differentiation.

For example, we know that d/dx (x²) = 2x
==> d(x²) = 2x dx,
==> 2x dx = x².
An important point to be noted here is that the symbol of an integral is " ".

So we can say that the test of integration is to differentiate the answer. If the integration is correct, the differential of the answer must equal the expression to be integrated.