intégrale

The integral is a fundamental concept in calculus, representing the area under a curve or the accumulation of quantities. It is denoted by the integral sign, which resembles a stylized "S" for "sum." The integral of a function f(x) with respect to x is written as ∫f(x) dx. This notation indicates that the integral is the limit of a sum of areas of rectangles under the curve as the width of the rectangles approaches zero.

There are two main types of integrals: definite and indefinite. An indefinite integral, also known as an antiderivative, is a function whose derivative is the original function. It is represented by the integral sign without limits, such as ∫f(x) dx. A definite integral, on the other hand, calculates the exact area under the curve between two points, a and b, and is written as ∫ from a to b f(x) dx. This is evaluated by finding the antiderivative of f(x) and then subtracting the value of the antiderivative at a from its value at b.

The integral has numerous applications in mathematics and physics, including calculating areas, volumes, and probabilities. It is also used to solve differential equations, which are equations involving derivatives. The fundamental theorem of calculus connects differentiation and integration, stating that the derivative of an integral of a function is the original function itself. This theorem is crucial for evaluating definite integrals and understanding the relationship between the two concepts.