The NCERT exercise 7.2 class 12 solutions are incredibly beneficial to students preparing for their class 12 exams. This paper contains solutions to chapter integration, which is part of chapter exercise 7.2 in class 12 Maths. Here you’ll find detailed answers to the chapter’s questions. An integral is a mathematical notion that assigns values to functions in order to explain the area, volume, displacement, and all other ideas that come from combining infinitesimal data. These solutions will assist you in learning all of these concepts and putting them into practice.
NCERT Solutions for Class 12 Maths Chapter 7 – Exercise 7.2
Introduction
One of the most important operations in calculus is integration, and the other is differentiation. One of the additive constants used to distinguish between characters is the integration operation. It is the calculus fundamental theorem that connects differentiation with definite integral.
In the late 17th century, the famous Isaac Newton and another prominent scholar Gottfried Wilhelm Leibniz separately devised the concepts of integration, thinking of integral as an endless sum of rectangles of infinitesimal width. Later, Bernhard Riemann developed a mathematical definition of integrals that is based on a limiting approach that approximates the entire area of a curved region by splitting it into thin vertical slabs separately.
Class 12 Maths NCERT solutions chapter 7 exercise 7.2 will teach you that the beginning of the 19th century was when more sophisticated conceptions of integrals began to appear. When, where, the type of function, and the domain across which the integration is done were all generalized during this time. A line integral is a function of variables that are covered (2 or more in number). A curve connects the two endpoints and takes the place of the integration interval. A piece of surface replaces the curve when the surface integral is applied in a 3D space.
Formalization
Despite the fact that Newton and Leibniz devised a methodical approach to integration, it had some flaws in terms of rigor. Bishop Berkeley was hostile to Newton’s notion, which he dubbed “ghosts of departed quantities.” Calculus was more widely recognized when it came to the development of limits. Riemann was the first to formalize integration with limits in 1854.
On a bounded interval, all bounded piecewise continuous functions are Riemann-integrable, but more generic functions—particularly in the context of Fourier analysis—were also considered, to which Riemann’s definition does not apply. Lebesgue devised a new definition of integral that differed from previous definitions and incorporated the concept of measure theory, which covered real or subfield analysis.
FAQ
Question. What are the three most popular and helpful integration methods in calculus?
Answer. The following are the three most common ways of integration:
- Method of substitution
- The Method of Partial Fractions
- Parts-based integration
Problems of integration of various forms are covered in class 12 maths chapter 7 ex 7.2. Each approach has its own set of principles that must be properly understood before the integration can be completed.
Question. What are the various uses for definite integrals?
Answer. In arithmetic, science, and economics, integration is used to determine the worth of a variety of objects.
The following are some of the most important applications of definite integrals:
- The circumference of a curve-bound region
- The volume of a solid whose cross-section has been determined.
- Arc length is the measurement of how long an arc is.
- Fluids’ pressure
- Integration and electric circuits
- Moment of inertia is a term that refers to the inertia
Question. Is Maths in Class 12 difficult?
Answer. Many students despise math and find Class 12 math to be extremely difficult. This, however, may not be the case. Everything is up to you and your plan. You will not be able to do well if you have a negative outlook. You must practice Class 12 Maths on a daily basis, and Infinity Learn’s NCERT Solutions Class 12 Maths is the ideal friend that can make things easier for you.
