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Comparison between Integration and Differentiation

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Differentiation and Integration both are main roots of mathematics and even mathematics cannot be imagined without differentiation and integration. Many important branches of mathematics are based on these two terms. After understanding these topics you will enjoy the numerical problems related to the differentiation and integration.

Differentiation

Differentiation is the method of finding the derivative of a function. The reverse method is termed as anti-differentiation.

According to the Calculus, the derivative can be stated as how a function changes as its input changes. A derivative can be explained as how much one quantity is changed  in response to changes in some another quantity.

Integration:

The opposite process is called antidifferentiation. According to Calculus the anti differentiation is known as integration.

Now we will see the Comparison between Integration and Differentiation

1. These two terms are related to a branch of Basic Mathematics which is called as Calculus. Integration is Adding up while Differentiation is dividing.

2. Integration tells about the distance travelled by the function while differentiation tells about the speed of the function.

3. We also can sat that Integral is equal to distance function is equal to  position derivative is equal to  speed

4. Differentiation makes large distance to very small Fractions while Integration makes the little fractions to large.

5. These both are opposite of each other. 
   Differentiation of cos is –sin while integration of cos is sin.

6. These are opposite’s terms of each other.

 If we take the differentiation of a function, then take the integration of the answer, we will complete it with the function again.

7. According to the definition, the differentiation of a function is the Slope of a function while the integration is the area between the function and the x-axis.

8. In a simple way we can say that differentiation is the means  of finding the rate of change of the incline/grade of any function while integration may be defined as  the area which is under the curve of function with respect to the x axis.

 

9. We can compare integration and differentiation by explaining their applications:

Integration can also be known as anti-differentiation. It can also be applied to determine the length of a line on a graph which is also termed as a path of a function and  the area of three dimensional graph Functions and also volumes of three dimensional graph Functions.

While if a Point is a local maximum or minimum of a function, frequent differentiation are able to  determine it and also can  be used in equations to help the explaining  the movement of substance.

10. Integration is the reverse of differentiation:

In one way it can be taken as differentiation in reverse.

Let’s take a example:

Suppose we will have to differentiate the function

 y = x4.

We obtain
dy/dx = 4x3.

In Integration we reverse this process and we can say that the integration of 4x3 is x4.

11. In simple way we can say that differentiation is applied to determine the rates of change of variables and integration is used to determine lengths, volumes  and areas.

This all can be explained by solving numerical examples:

Let take  a example
z = 3x2 +6x
Now  means of differentiation is (dy/dx) rate of change of z for a very minute change of x.

We can do differentiation in many steps:

Firstly we multiply the variable 3 by power 2 and power 2 will be decreased by one. It will be resulted in 6x.

In case of 6x, 6 will be multiplied by power 1 and power 1 will be decreased by one. It will  be resulted in 6

 Here:

The differentiation of  z = 3x2 +6x is

dz /dx =6x + 6


Again suppose:

 it is given that 
dy/dx = 6x + 6

Now we are to determine the function from which this differentiation has been derived. The following process by which we can go back to that function is known as the integration.

Integration of 6x+6

We can do integration in many steps:

Firstly we increase the power of variable by one and divide this variable by increased power.

In case of 6x power of x is given as one. We will increase it by one that will be 2 and after dividing by 2 it will result in 6x.

In case of 6 the power of x is 0. We will increase it by one that will be 1 and after dividing by 1 it will result in 3x2.

Here:

Then integration of 6x+6 will be as 3x2 +6x.

So, in this way we find difference between integration and differentiation.

 

Relation between Integration and Differentiation

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Mathematical Calculus can be divided into two parts that is Differentiation calculus and Integration calculus and both are related with each other. First of all we discuss what is differentiation? Differentiation is a process to calculate rate or change like, if we want to calculate rate of function f(x), then we use differentiation and we represent differentiation by dy/dx where ‘y’ represents function f(x) in above case. Now we discuss what is integration? Integration is a process to calculate collection of data means if we want to calculate the collection of data in function f(x), we use integration and we represent integration by ∫ y dx where ‘y’ is a function f(x) in above case.
Now we discuss relationship between differentiation and integration:
Integration and differentiation both are opposite to each other means if we want to calculate collection, we use integration and if we want to calculate single quantity, we use differentiation. Now we take an example to understand the relationship between differentiation and integration -
Example: Let we have a function f(x) = x and when we differentiate this function with respect to ‘x’, then -
d/dx[f(x)] = d/dx(x) = 1
and when we integrate this result, which is provided by differentiation, then -
∫ (d/dx[f(x)]) * dx = ∫ 1 dx = x = f(x).
So, result of integration provides the original function again. Therefore we can say that differentiation and integration are opposite to each other and we can write relationship between them in following manner -
∫ (d/dx[f(x)]) * dx = d/dx ( ∫ f(x) dx) = f(x).
This is a one example which tells relationship between differentiation and integration, we can take many more example to prove Relation between Integration and Differentiation like, differentiation of sin x provide cos x as a result and when we integrate cos x, it provide sin x again. Therefore we can say that integration process is a reverse process of differentiation. This is the relation between Integration and Differentiation.