Terminology and Notation
The simplest case, the integral over x of a real-valued function f(x), is written as
The integral sign ∫ represents integration. The dx indicates that we are integrating over x; dx is called the variable of integration. In correct mathematical typography, the dx is separated from the integrand by a space (as shown). Some authors use an upright d (that is, dx instead of dx). Inside the ∫...dx is the expression to be integrated, called the integrand. In this case the integrand is the function f(x). Because there is no domain specified, the integral is called an indefinite integral.
When integrating over a specified domain, we speak of a definite integral. Integrating over a domain D is written as
- or if the domain is an interval of x;
The domain D or the interval is called the domain of integration.
If a function has an integral, it is said to be integrable. In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense (such as a sample space in probability theory).
In the modern Arabic mathematical notation, which aims at pre-university levels of education in the Arab world and is written from right to left, a reflected integral symbol is used (W3C 2006).
The variable of integration dx has different interpretations depending on the theory being used. It can be seen as strictly a notation indicating that x is a dummy variable of integration; if the integral is seen as a Riemann sum, dx is a reflection of the weights or widths d of the intervals of x; in Lebesgue integration and its extensions, dx is a measure; in non-standard analysis, it is an infinitesimal; or it can be seen as an independent mathematical quantity, a differential form. More complicated cases may vary the notation slightly. In Leibniz's notation, dx is interpreted an infinitesimal change in x, but his interpretation lacks rigour in the end. Nonetheless Leibniz's notation is the most common one today; and as few people are in need of full rigour, even his interpretation is still used in many settings.
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