What is the difference between definite and indefinite integrals?

Definite and indefinite integrals are two fundamental concepts in calculus. Here's how they differ:

1. Purpose: The indefinite integral is essentially the antiderivative of a function. It represents a family of functions. On the other hand, the definite integral is used to calculate the area under the curve of a function, from one point to another.

2. Notation: The indefinite integral of a function f(x) is denoted as ∫f(x) dx, while the definite integral is denoted as ∫ from a to b f(x) dx, where a and b are the limits of integration.

3. Result: The indefinite integral of a function is another function (plus the constant of integration, C). However, the definite integral of a function is a number, which represents the signed area between the function and the x-axis, within the limits of integration.

4. Fundamental Theorem of Calculus: The relationship between definite and indefinite integrals is given by the Fundamental Theorem of Calculus. It states that if F is an antiderivative of f on an interval, then the definite integral of f from a to b is equal to F(b) - F(a).

In summary, while both definite and indefinite integrals are key tools in calculus, they serve different purposes and yield different types of results.