The integral is one of the two base concepts of calculus, along with the derivative.
In introductory, high-school level texts, the integral is often presented in two parts, the indefinite integral and definite integral. Although this approach lacks mathematical formality, it has the advantage of being easy to grasp and convenient to use in most of its applications, especially in Physics.
The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function is written as , then the indefinite integral of is , where is a real constant. This is because the derivative of a constant is .
- The integral of a function is written as , where the means that the function is being integrated in relation to .
- Often, to save space, the integral of is written as , the integral of as , etc.
Rules of Indefinite Integrals
- for a constant and another constant .
The definite integral is also the area under a curve between two points and . For example, the area under the curve between and is , as area below the x-axis is taken as negative area.
Definition and Notation
- The definite integral of a function between and is written as .
- , where is the antiderivative of . This is also notated , read as "The integral of evaluated at and ." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
Rules of Definite Integrals
- for any .
The notion of an integral is one of the key ideas in severel areas of higher mathematics including analysis and topology. The integral can be defined in several ways which can be applied to several different settings. However, the most common definition, and the one which most closely resembles the the 'definite integral' is the Riemann Integral
We say that is Riemann Integrable on if and only if
such that if is a tagged partition on with , where is the Riemann sum of with respect to
is said to be the integral of on and is written as
Another integral commonly used in introductory texts is the Darboux Integral (which is often called the Riemann Integral)
We say that is Darboux Integrable on if and only if , where and are respectively the lower sum and upper sum of with respect to partition
The notation used for the Darboux integral is the same as that for the Riemann integral.
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Other important definitions of integration include the Riemann-Stieltjes integral, Lebesgue integral, Henstock-Kurzweil integral, etc.
- The word integral is the adjectival form of the noun "integer." Thus, is integral while is not.