In mathematics, an “integral” has two primary meanings:
- Definite Integral:
- The definite integral is a mathematical concept used in calculus. It represents the signed area under the curve of a function over a specified interval. The definite integral of a function f(x) from a to b is denoted by ∫abf(x)dx. Geometrically, this integral corresponds to the area between the curve and the x-axis over the given interval.
- Indefinite Integral:
- The indefinite integral is also a concept in calculus. It represents the antiderivative of a function, which is a function whose derivative is the original function. The indefinite integral of a function f(x) is denoted by ∫f(x)dx. The symbol ∫∫ represents the integration process, and dx indicates the variable of integration.
Mathematically, the relationship between the definite and indefinite integrals is expressed by the Fundamental Theorem of Calculus. The theorem states that if F(x) is an antiderivative of f(x), then ∫abf(x)dx=F(b)−F(a). In other words, the definite integral can be evaluated by finding the difference between the values of the antiderivative at the upper and lower limits of integration.
The integral is a fundamental concept in calculus and plays a crucial role in various mathematical applications, including determining areas under curves, calculating volumes of solids of revolution, and solving differential equations.