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Learn Integration in detail with its types and methods
Learn about integration in detail with all its types and methods to know the importance of integration
What is Integration?
The process of summing up minute sections of a whole area is known as integration. The integration is yet 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 reduces in its parts while in integration we add up the parts of a function. Due to this reason in mathematics integration use to determine the summation beneath an extensive area.
Mathematical definition
The integration can define 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 extra calculations is an easy task that may be done either or with the help 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 divide into two distinct types that both are of equal significance in maths.
Definite Integrals
The integrals comprising both the upper and lower limits of function are regarded as the definite integrals or the Reiman’s integral. The definite integrals calculate using the given expression
∫baf(x)dx
The b and a in the above formula are the upper and lower limits of the function respectively.
Indefinite Integrals
The integral that represent without upper and lower bound or doesn’t have upper and lower limits of a function is the indefinite integral. The indefinite integrals are represent 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 is use for the process of antidifferentiation.
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Integration by Substitution
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Integration by Part
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Partial fraction
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Trigonometric Integrals
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Trigonometric Substitution
Methods of Integration:
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Shell Method
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Disc Method
Shell Method:
Shell method is for when integrating along an axis “perpendicular” to the axis of revolution. This is in contrast to disc integration which integrates along the axis “parallel” to the axis of revolution.
Disc Method:
It is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis “parallel” to the axis of revolution.
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 Shell Method calculator with steps or this Disc Method calculator with steps. The choice is yours!!
Substitution Method
A function can integrate by substitution method, in which u is as the function of x and solve incorporating the formula
u’ = du/dx
This expression is further represented in terms of integration as given below, the u is substitute by g(x).
∫ f(u)u’ dx = ∫ f(u)du
By Parts
The integration by parts technique is when the two functions are in the form of a product. The formula used to integrate two functions by parts method is represent 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 implied solving various mathematical problems that were nearly impossible before Newton and Leibniz proposed it. Some major concepts that simplified using integrals and represent the importance of integration in mathematics include the following:
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Areas Under the Curve
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Arc Length of a Curve
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Surface Area of a Curve
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The Area between two Curves
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Finding Volume of Solids
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The volume of Revolution – Cylindrical Shells
Area between two Curves
Just as the area under a curve s calculated similarly using the definite integrals, the area between two curves, calculated by subtracting the bottom curve from the top one in cases when the top one is higher than the lower curve.
Moreover, to determine the limits of integration we may need to locate intersection points of the two curves. If the functions cross each other in the integration interval, we then have to divide the integral.
The volume of the Solids
The volume of the solid objects that exist in three dimensions is determined using the definite integrals. The integration involves several different techniques to calculate the volume of solids based on solid’s characteristics.
However, by either implying the slicing, washer or disk method the volume of solids could determine.
Surface Area and Arc Length of Curves
The integrals can use to compute the arc length of a curve. This achieves by taking an initial approximation of the arc length made by utilizing line segments, which results in a Riemann sum.
The formula for definite integrals can then be calculated by taking a limit. Similarly, a surface of revolution may be found by using the same ideas that were used to determine arc length. However, it’s typically difficult to assess both arc length and surface area formula integrals.
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