Flashcards: Applications of Integrals | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
INTEGRATION 
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as 
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate 
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc. 
whenever the ratio of change of a quantity is known we can always calculate the total change in a 
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments 
of force etc. are some of the innumerable applications of integration. 
  
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INTEGRATION 
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as 
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate 
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc. 
whenever the ratio of change of a quantity is known we can always calculate the total change in a 
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments 
of force etc. are some of the innumerable applications of integration. 
  
 
 
 
 
INTEGRALS AS AN ANTIDERIVATIVE 
The process of finding the antiderivative is called integration. 
Integration is the inverse process of differentiation. 
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ?? 
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? ) 
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? ) 
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand. 
So ??? (?? )???? is not unique due to presence of ' ?? '. 
  
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INTEGRATION 
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as 
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate 
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc. 
whenever the ratio of change of a quantity is known we can always calculate the total change in a 
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments 
of force etc. are some of the innumerable applications of integration. 
  
 
 
 
 
INTEGRALS AS AN ANTIDERIVATIVE 
The process of finding the antiderivative is called integration. 
Integration is the inverse process of differentiation. 
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ?? 
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? ) 
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? ) 
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand. 
So ??? (?? )???? is not unique due to presence of ' ?? '. 
  
 
 
 
 
Geometrical Significance 
The derivative of a function give the slope whereas, the integration of a bounded function gives a 
particular algebraic area. 
  
Page 4


 
 
 
INTEGRATION 
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as 
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate 
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc. 
whenever the ratio of change of a quantity is known we can always calculate the total change in a 
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments 
of force etc. are some of the innumerable applications of integration. 
  
 
 
 
 
INTEGRALS AS AN ANTIDERIVATIVE 
The process of finding the antiderivative is called integration. 
Integration is the inverse process of differentiation. 
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ?? 
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? ) 
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? ) 
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand. 
So ??? (?? )???? is not unique due to presence of ' ?? '. 
  
 
 
 
 
Geometrical Significance 
The derivative of a function give the slope whereas, the integration of a bounded function gives a 
particular algebraic area. 
  
 
 
 
Derivative of Integral, Integral of Derivative 
?? ????
? ?? (?? )???? = ?? (?? )
 ? {
?? ????
(?? (?? ))}???? = ?? (?? ) + ?? ? { ?? (?? )?? '
(?? ) + ?? '
(?? )?? (?? )} ???? = ?? (?? ) · ?? (?? ) + ?? ? {
?? '
(?? ) · ?? (?? ) - ?? (?? )?? '
(?? )
{?? (?? )}
2
} ???? =
?? (?? )
?? (?? )
+ ?? ? ?? '
(?? (?? ))?? '
(?? )???? = ?????? (?? ) + ?? 
  
Page 5


 
 
 
INTEGRATION 
The word 'Integration' literally means 'Summation'. It is applied to almost all branches of science such as 
Physics, Chemistry, Biology, Engineering, Economics, Statistics, Algebra, trigonometry, Coordinate 
geometry, Geometry dynamics, and even to Social sciences, Quality control, pediatrics, Mensuration etc. 
whenever the ratio of change of a quantity is known we can always calculate the total change in a 
specified interval of time. Finding areas, volumes, lengths, moment of inertia, pressure, work, moments 
of force etc. are some of the innumerable applications of integration. 
  
 
 
 
 
INTEGRALS AS AN ANTIDERIVATIVE 
The process of finding the antiderivative is called integration. 
Integration is the inverse process of differentiation. 
If ?? '
(?? ) = ?? (?? ) then ??? (?? )???? = ?? (?? ) + ?? 
i.e. ??? (?? )???? = ?? (?? ) if ?? '
(?? ) = ?? (?? ) 
??? (?? )???? = ?? (?? ) + ?? as (?? (?? ) + ?? )
'
= ?? (?? ) 
Here ' ?? ' is called constant of integration and ?? (?? ) is called integrand. 
So ??? (?? )???? is not unique due to presence of ' ?? '. 
  
 
 
 
 
Geometrical Significance 
The derivative of a function give the slope whereas, the integration of a bounded function gives a 
particular algebraic area. 
  
 
 
 
Derivative of Integral, Integral of Derivative 
?? ????
? ?? (?? )???? = ?? (?? )
 ? {
?? ????
(?? (?? ))}???? = ?? (?? ) + ?? ? { ?? (?? )?? '
(?? ) + ?? '
(?? )?? (?? )} ???? = ?? (?? ) · ?? (?? ) + ?? ? {
?? '
(?? ) · ?? (?? ) - ?? (?? )?? '
(?? )
{?? (?? )}
2
} ???? =
?? (?? )
?? (?? )
+ ?? ? ?? '
(?? (?? ))?? '
(?? )???? = ?????? (?? ) + ?? 
  
 
 
 
Fundamental rules of Integration 
 ? { ?? 1
(?? ) ± ?? 2
(?? ) ± ?? 3
(?? ) ± ? ± ?? ?? (?? )} ???? = ? ?? 1
(?? )???? ± ? ?? 2
(?? )???? ± ? ?? 3
(?? )???? ± ? ± ? ?? ?? (?? )????
 ? ???? (?? )???? = ?? ? ?? (?? )???? , where ?? is a constant (?? ? 0)
 ? ?? '
(???? + ?? )???? =
?? (???? + ?? )
?? + ?? (?? ? 0)