itégration

Intégration is a fundamental concept in mathematics, referring to the process of finding the integral of a function. An integral can be thought of as the antiderivative of a function, meaning a function whose derivative is the original function. This process is the inverse of differentiation.

There are two main types of integrals: indefinite and definite. An indefinite integral, often denoted by ∫f(x) dx, results in a family of functions that differ by a constant, known as the constant of integration. This constant arises because the derivative of a constant is zero.

A definite integral, denoted by ∫[a to b] f(x) dx, represents the net area under the curve of the function f(x) between the limits a and b. The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration, stating that the definite integral of a function can be evaluated by finding its antiderivative and then subtracting the value of the antiderivative at the lower limit from its value at the upper limit.

Integration has wide-ranging applications in various fields, including physics (calculating work, displacement, and velocity), engineering (determining volumes, centroids, and moments of inertia), economics (modeling cumulative effects), and statistics (calculating probabilities). It is a cornerstone of calculus and a powerful tool for solving problems involving continuous change.