Integration - Demystified

Integration is something closely associated with differentiation (click here to know what is differentiation). We all have studied lots of standard integrals, but do we know the answer to following questions.
What is integration?
Why is integration needed?
What are some practical examples of integration?
If you know the answer then you need not read this post, if you don't know the answer reading this is sure to help you.

What is integration?
The process of finding the value of an integral is called integration.
Well am I supposed to repeat this standard definition. No, let us try to understand it in a different manner.
Consider the following graph.

A rectangle is shown inside a graph whose properties are as given below
Length = 5Breadth = 4
Area of this rectangle = 5 x 4 = 20.
Let me explain this rectangle little elaborately. The length of the rectangle is shown along the X axis. For all the values of X, the value of Y is a constant. ie when X varies from 0 to 5, the value of Y is 4. Hence we could calculate the area by simply multiplying.
If the value of Y was changing then the figure won't be a rectangle.
Consider the figure given below

Is this a rectangle? No..
Why? The value of Y is not a constant. As X is varying the value of Y is varying as shown below

X = 1, Y = 4

X = 2, Y = 3

X = 3, Y = 2

X = 4, Y = 1
Now how can we calculate the area of this shape. We are trying to find out the area of the shaded region.
If the figure is broken down as shown below we can find the area

Each of the above figure is a rectangle. So the area of the total shape is the area of each of the shapes given above. Since each of the above figure is a rectangle calculating area of each of the shapes is simple.
Area = (L1 x B1) + (L2 x B2) + (L3 x B3) + (L4 x B4)

Area = 1 x 4 + 2 x 3 + 3 x 2 + 4 x 1 = 20
Now we can analyse how we found the area of this irregular shape.
For each value of X we found that the value of Y is remaining a constant. Hence we were able to calculate the area for each value of X and then finally add up the results together and find the total area
If you understood the above statement, you understood what is integration.

Integration is the technique used to find the area under any shape.

How to find the area under any shape?

Just like the way we found out the area of the last shape. For each value of X where Y is a constant, find the area and then sum up the values. Here we used the principle of integration to find out the area.

Now we can explain these things in a more mathematical terminology.

If a variable "y" is changing as the value of "x" is increasing in a particular pattern, then we say that y is a function of xie y = f(x)

So the shapes which we showed above also can be written as y = f(x)

To calculate the integral of y,

means to calculate the integral of f(x)

means to find out the area under the curve when 'x' varies from 1 to 4 and 'y' varies from 1 to 4

The steps are

1. Divide the total shape in small intervals of X. This small interval of X is called "dx".
2. Find out the area. Area = breadth x length, here breadth = y, (value of y at that point), hence area = y. dx
3. Take the sum of all areas corresponding to each dx.the symbol for integral is an elongated "S" which represents sum

So integration is a method for finding the area under a curve, where the equation of the curve is given by y = f(x)

Why do we need to study the standard integration of various functions?

In the last example we calculated the area of a figure when the value of x was varying from 1 to 4. Also the value of y contianed 4 different values. Hence manual computation was possible.

If there were X was needed to be divided into 1000 values, then we have to find the area of 1000 different shapes and then find the sum. Will it be possible to manually compute this.
So what will we do?

The curve can be represented as y = f(x), where f(x) may be some standard function.

So if the integral of the standard function is known we can compute the area very simply.
Consider the function y = x

Here if we are asked to find out the integral of y, when x varies from 0 to 4, we can manually calculate like the following

the function is y = x.
hence when

x = 0, y = 0

x = 1, y = 1

x = 2, y = 2

x = 3, y = 3

x = 4, y = 4
The figure can be drawn as shown below.

To manually compute the area, we can find out the number of squares in the shaded region,
there are 6 squares and 4 half squares
each square's area is 1.
hence the total area is 6 + 2 = 8.
If we use integration

Using integration also we got the same result.

If we need to find the integral of y = x, when x varies from 0 to 100, can we do it manually?
It is quite tedious, to count the number of squares like we did earlier.
But if we use the integration equation then we can compute as follows

Whatever complex the curve is, if we can express it as a function of x, then to find the area we need to compute the expression after integration, then substitute the limit values to get the result.

What is the practical application of Integration?

This can be used to find the area of a plane what ever be its shapeExtending the principle this can be used to find the volume of a 3D shape.Finding area and volume has n number of practical applications.

Hope you understood the concepts. Now plunge into one of those text books and get more detail. It will become more interesting :)