Flashcards: Applications of Integrals

``` 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.

Page 2

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 ' ?? '.

Page 3

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)

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## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Flashcards: Applications of Integrals - Mathematics (Maths) for JEE Main & Advanced

 1. What are some real-world applications of integrals in JEE?
Ans. Some real-world applications of integrals in JEE include calculating areas under curves, finding the volume of solids of revolution, determining the work done by a force, and finding the center of mass of a system.
 2. How can integrals be used to calculate the area between two curves in JEE?
Ans. To calculate the area between two curves in JEE, you can find the points of intersection of the curves, set up the integral with the upper and lower curves as the limits of integration, and then integrate the difference between the two curves with respect to the variable.
 3. In JEE, how is the concept of integrals applied to find the average value of a function over an interval?
Ans. In JEE, to find the average value of a function over an interval, you can use the formula (1/(b-a)) * ∫[a,b] f(x) dx, where 'a' and 'b' are the limits of the interval and f(x) is the function.
 4. How can integrals be used to calculate the total distance traveled by an object in JEE?
Ans. In JEE, to calculate the total distance traveled by an object, you can find the absolute value of the velocity function, set up the integral of the velocity function over the given time interval, and then integrate to find the total distance traveled.
 5. What is the significance of integrals in JEE when determining the mass of an object with varying density?
Ans. In JEE, integrals are used to determine the mass of an object with varying density by setting up an integral with the density function as the integrand and the volume element as the variable of integration. By integrating over the volume of the object, you can find the total mass.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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