The integral is one of the main operators in calculus, which is usually split into derivative calculus and integral calculus. Visually the integral of a function can be interpreted as the area under the curve between two points, where the area underneath the x-axis is considered negative.
The definite integral calculates the accumulated sum (area) under the curve of a function over a specific interval [a, b]. It can be denoted as ∫[a, b] f(x) dx. The definite integral results in a numerical value, which represents the net area between the function’s curve and the x-axis. This area is considered positive when the function lies above the x-axis and negative when it lies below.
The indefinite integral, also known as an antiderivative, represents the family of functions that have the given function as their derivative. It does not have specific limits and is denoted as . The result of an indefinite integral is a function, plus an arbitrary constant , since taking the derivative of a constant results in zero. The indefinite integral provides a general expression for calculating the area under the curve of the function over any interval.