What is Integration
Mathematically, the process of summing up extremely minute sections of a whole area is known as integration. The integration is however also known as antidifferentiation as it is the inverse process of differentiation.
Technically, in differentiation or the process of finding the derivative of a function, a function is reduced in its parts while in integration we add up the parts of a function. Due to this reason in mathematics integration is used to determine the summation beneath an extensive area.
The integration can be defined in terms of mathematics by stating, the sum of the rectangles to the curve at each interval of change in x as the number of rectangles increases to infinity is the integral of the function of x from a to b.
Integration in Maths
Making minor additional calculations is an easy task that may be done either manually or with the assistance of calculators, depending on the complexity of the problem.
On the other hand, integration methods are used in mathematics for big addition problems where the boundaries may stretch to infinity or beyond. Integration along with differentiation have a huge role to play in maths that’s why forms the fundamental theorem of calculus.
Types of Integration in Maths
The integration is divided into two distinct types that both are of equal significance in maths.
The integrals comprising both the upper and lower limits of a function are regarded as the definite integrals or the Reiman’s integral. The definite integrals are calculated using the given expression
The b and a in the above formula are the upper and lower limits of the function respectively.
The integral that is represented without upper and lower bound or doesn’t have upper and lower limits of a function is called the indefinite integral. The indefinite integrals are represented by the given formula
∫f(x)dx = F(x) + C
How to do Integration
The integration is a vast method of summing up small portions infinitely and comprises several different techniques to integrate a function. The following techniques of integration are used for the process of antidifferentiation.
- Integration by Substitution
- Integration by Part
- Integration by Partial fraction
- Integration by Trigonometric Integrals
- Integration by Trigonometric Substitution
Types of Integrals:
- Double integrals
- Triple integrals
Double integration is for integrating over two dimensional regions while triple integration is for integrating over three dimensions.
You can also use online tools that are quite advanced for doing all of your integration-related problems step by step. Such as you can see this Triple integral calculator with steps. The choice is yours!!
A function can be integrated by substitution method, in which u is given as the function of x and is solved incorporating the formula
u’ = du/dx
This expression is further represented in terms of integration as given below, the u is substituted by g(x).
∫ f(u)u’ dx = ∫ f(u)du
The integration by parts technique is implied when the two functions are in the form of a product. The formula used in order to integrate two function by parts method is represented as
∫f(x)g(x) dx = f(x)∫ g(x) dx – ∫ (f'(x) ∫g(x) dx) dx
By Partial Fraction
Using the partial fractions technique, we may integrate rational algebraic functions whose numerator and denominator both include positive integral powers of x with constant coefficients.
You must first decompose the improper rational function into its proper rational function and then integrate the result to obtain ∫ f(x)/g(x) dx.
And therefore, can denote it as
∫f(x)/g(x) dx = ∫ p(x)/q(x) + ∫ r(x)/s(x), where g(x) = a(x) . s(x)
Importance of Integration in Mathematics
The concept of integration is implied to solve various mathematical problems that were nearly impossible before Newton and Leibniz proposed it. Some major concepts that are simplified using integrals and represent the importance of integration in mathematics include the following:
- Areas Under the Curve
- Arc Length of a Curve
- Surface Area of a Curve
- The Area between two Curves
- Finding Volume of Solids
- The volume of Revolution – Cylindrical Shells