Definite integration is the process of finding the area enclosed between the curve and the x-axis over a specific interval. The Class 12 Mathematics syllabus covers the following topics related to definite integration:
Definite Integral as a Limit of a Sum: This topic covers the concept of definite integral as a limit of a sum and its notation.
Fundamental Theorem of Calculus: This topic covers the fundamental theorem of calculus, which states that integration and differentiation are inverse processes of each other.
Properties of Definite Integrals: This topic covers the properties of definite integrals, such as linearity, additivity, and symmetry.
Evaluation of Definite Integrals: This topic covers the various techniques for evaluating definite integrals, such as substitution, integration by parts, and partial fractions.
Definite Integrals as a Limiting Process: This topic covers the concept of definite integrals as a limiting process and its relation to the Riemann sum.
Definite integration is important because it provides a powerful tool for calculating areas, volumes, and solving differential equations. The fundamental theorem of calculus is used to evaluate definite integrals, and the properties of definite integrals are used in solving complex problems related to optimization, related rates, and curve sketching. The techniques for evaluating definite integrals are used in solving problems related to physics, engineering, economics, and statistics. The concept of definite integrals as a limiting process is used in proving results related to the continuity and differentiability of functions.