Integral

 An integral is a fundamental concept in calculus, a branch of mathematics that deals with the study of change and motion. The integral is used to determine the area under a curve or to find the accumulation of a quantity over an interval. It is the reverse process of differentiation.


There are two main types of integrals: the definite integral and the indefinite integral.


Definite Integral:

The definite integral of a function f(x) over an interval [a, b] represents the area between the curve of the function and the x-axis within that interval. It is denoted by ∫[a, b] f(x) dx. The result of the definite integral gives a numeric value.

Example: ∫[0, 2] x^2 dx represents the area under the curve y = x^2 between x = 0 and x = 2.


Indefinite Integral:

The indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the original function. It is denoted by ∫f(x) dx, and it includes a constant of integration (C) because the antiderivative is not unique.

Example: ∫x^2 dx = (1/3)x^3 + C, where C is the constant of integration.


Integration is a powerful tool used in various fields such as physics, engineering, economics, and many other areas to solve problems involving rates of change, areas, volumes, and more. It is a crucial concept for understanding the behavior of functions and analyzing real-world scenarios.

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