An overview of The Integrals Chapter
Integrals are mathematical objects, which can be interpreted as a generalisation of an area. Moreover, coupled with derivatives, they form the fundamental of calculus. Integrals are divided into two subtypes, definite integrals and indefinite integrals.
Additionally, every definition of integrals has a perfect measure. It is the base of integration. For instance, Riemann integral is based on Jordan Measure. Similarly, the Lebesgue integral is based on Lebesgue measure. Moreover, the Riemann integral is regarded as the simplest integral definition and used in physics and calculus.
Math class 12 chapter 7 integrals
In this chapter, you will study about integrals and their applications.
7.1- Introduction
At the beginning of the chapter, you are introduced to the concept of integration meaning, and its history. Moreover, you also learn about definite and indefinite integrals. Furthermore, you are introduced to the fundamental concept of indefinite integrals. Integration is also regarded as anti-derivative.
7.2- Integration As An Inverse Process Definition
In this section of integrals, you will learn how to use the inverse process of differentiation. Instead of differentiating a function, you will get a derivative of a particular function, after that, you need to find the primitive function. This process of calculating integrals is regarded as the inverse process. There are several integration examples to help you comprehend the topic even better.
7.3- Methods of Integration
You have already learnt about the integrals functions of which are readily obtained from derivatives. Additionally, it was based on inspection, i.e. searching functions which derivatives will lead to integrals. However, this section deals with a higher level of sums that require different methods like –
Integration via submission.
Integration by partial functions.
integration by parts.
Additionally, there is an exercise with different questions which require critical thinking for solving these problems.
7.4- Integrals of Some Particular Functions
As you have already studied about trigonometric functions like sine, tangent, etc. In this section, you use those functions to build formulas, which will help you during an exam.
7.5- Integration by Partial Fractions
The focus point of this section, disintegrating equations into parts and discovering the integral of every section. This partial disintegration process makes the overall approach easier. Additionally, it proves highly efficient in finding a solution to any problem.
7.6- Integration by Parts
This section is important for understanding the processes of solving problems. Moreover, in this portion, you learn more than one method of integration. Hence, these methods are useful in integrating products of functions. However, you should know that it does not apply to a product of functions in every case.
7.7- Definite Integral
In the previous sections, you have already covered indefinite integrals and methods to find them. Moreover, it also included discussion of some special integrals. In this section, you will learn about definite integrals.
Furthermore, definite integrals have a unique value. It is either introduced as the ‘limit of a sum’ or if it has an anti-derivative ‘F’ at its endpoints.
7.8- Fundamental Theorem of Calculus
This section of integrals is very interesting as it combines both mensuration and calculus. Therefore, it helps in finding areas covered under curves denoted by coordinates. Moreover, you get a chance to study fundamental theorems in this chapter.
7.9- Evolution of Definite Integrals by Substitution
Integration by substitution is a vital method to determine a definite integral of a function. You have already learnt about other methods in the previous sections. Moreover, substitution method also has a similar approach like others. However, you should practice every method.
7.10- Some Properties of Definite Integrals
In this section, you will learn about various essential properties of definite integrals. Furthermore, knowing these properties will help you evaluate definite integrals.
7.11- Exercise
In this exercise section, you get a chance to revise what you have studied till now. The questions are all based on concepts of integration.
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Integrals is an important chapter of class 12 mathematics. Moreover, it helps you to learn the fundamentals of this topic and then moves on to the advance sections. Additionally, you can visit the website as well as the app of Vedantu to access various study guides for this chapter.
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FAQs on Integrals
1. What is integration?
Ans. Integration is the opposite of differentiation. Moreover, in differential calculus, you get a function and need to find the derivative or differential of that function. However, in integral calculus, you have to find a function whose differential is given. Hence, integration is also known as inverse differentiation.
Furthermore, each definition of integrals has a seamless measure. It is the foundation of integration. For example, Riemann integral is based on Jordan Measure. Similarly, the Lebesgue integral is based on Lebesgue measure. Moreover, the Riemann integral is observed as the simplest integral definition and used in both physics and calculus.
2. What is calculus?
Ans. Calculus is a section of mathematics, which concentrates on varying calculations. Before the calculus, math was static. Therefore, it would only calculate objects that were perfectly still. However, this universe is continuously changing. Hence, no object is at rest, and they are always moving. Calculus helps to determine this movement and change of every particle.
Furthermore, calculus is used in fields like physics, economics, medicine, engineering, space travel, etc. Calculus was developed by two mathematicians Isaac Newton and Gottfried Leibniz. Newton first developed calculus and then applied it to understand various physical systems. On the other hand, Leibniz developed the notations used in calculus.
3. What are the practical applications of calculus?
Ans. There are several practical applications of calculus. Moreover, concepts like motion, heat, light, electricity, acoustics, harmonics, astronomy, etc. use calculus. Furthermore, calculus is used in geography, artificial intelligence, computers, photography, robotics, video games, etc.
Additionally, it assists in the calculations of radioactive decay in chemistry, predicts birth and death rate. Furthermore, it helps in studying gravity, planetary motion, geometric curves, civil engineering, flow of fluids, etc.
A prominent example is Einstein’s theory of relativity, which being a field of mathematics, helps companies to predict their profits.
4. What is the difference between differential and integral calculus?
Ans. There are two branches of calculus, differential and integral. Differential calculus governs the rate of change of a quantity. Moreover, it observes slopes and curves. Additionally, differential calculus deals with the rate of change of functions with respect to their variables. It especially uses differentials and derivative.
On the other hand, integral calculus tries to determine the quantity, where the rate of change is familiar. It primarily focuses on concepts like slopes, tangent lines, and velocities. Therefore, while differential calculus focuses on the curve, integral calculus highlights on the area under that curve.
