Problems related to Integration – This class is Polytechnic, 2nd Semester, Chapter 11 of Mathematics 2. We are regularly publishing all problem solving classes of Chapter 11 in “Gurukul Math”. Stay connected by subscribing.
Problems related to Integration
Integration Definition
The integration denotes the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly. In a broad sense, in calculus, the idea of limit is used where algebra and geometry are implemented. Limits help us in the study of the result of points on a graph such as how they get closer to each other until their distance is almost zero. We know that there are two major types of calculus –
- Differential Calculus
- Integral Calculus
The concept of integration has developed to solve the following types of problems:
- To find the problem function, when its derivatives are given.
- To find the area bounded by the graph of a function under certain constraints.
These two problems lead to the development of the concept called the “Integral Calculus”, which consist of definite and indefinite integral. In calculus, the concept of differentiating a function and integrating a function is linked using the theorem called the Fundamental Theorem of Calculus.
Maths Integration
In Maths, integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts. This method is used to find the summation under a vast scale. Calculation of small addition problems is an easy task which we can do manually or by using calculators as well.
But for big addition problems, where the limits could reach to even infinity, integration methods are used. Integration and differentiation both are important parts of calculus. The concept level of these topics is very high. Hence, it is introduced to us at higher secondary classes and then in engineering or higher education. To get an in-depth knowledge of integrals, read the complete article here.
Integral Calculus
According to Mathematician Bernhard Riemann,
“Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.” Learn more about Integral calculus here.
Let us now try to understand what does that mean:
- Take an example of a slope of a line in a graph to see what differential calculus is:
In general, we can find the slope by using the slope formula. But what if we are given to find an area of a curve? For a curve, the slope of the points varies, and it is then we need differential calculus to find the slope of a curve.
You must be familiar with finding out the derivative of a function using the rules of the derivative. Wasn’t it interesting? Now you are going to learn the other way round to find the original function using the rules in Integrating.
Integration – Inverse Process of Differentiation
We know that differentiation is the process of finding the derivative of the functions and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. So we can say that integration is the inverse process of differentiation or vice versa. The integration is also called the anti-differentiation. In this process, we are provided with the derivative of a function and asked to find out the function (i.e., primitive).
We know that the differentiation of sin x is cos x.
It is mathematically written as:
(d/dx) sinx = cos x …(1)
Here, cos x is the derivative of sin x. So, sin x is the antiderivative of the function cos x. Also, any real number “C” is considered as a constant function and the derivative of the constant function is zero.
So, equation (1) can be written as
(d/dx) (sinx + C)= cos x +0
(d/dx) (sinx + C)= cos x
Where “C” is the arbitrary constant or constant of integration.
Generally, we can write the function as follow:
(d/dx) [F(x)+C] = f(x), where x belongs to the interval I.
To represent the antiderivative of “f”, the integral symbol “∫” symbol is introduced. The antiderivative of the function is represented as ∫ f(x) dx. This can also be read as the indefinite integral of the function “f” with respect to x.
Therefore, the symbolic representation of the antiderivative of a function (Integration) is:
y = ∫ f(x) dx
∫ f(x) dx = F(x) + C.
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