definite integral

of: a function defined on an interval from to

is denoted:

where the values and are known as the lower and upper limits of integration, is called the integrand, and the symbol is the element of integration which shows that is the integration variable with respect to which the integration is to be performed. [M5.1, M5.2, P2.4]

is defined: by the limit of a sum:

with

where the sequence of values is such that and is the largest of the [M5.1, M5.2, P2.4]

may be interpreted: for a given function between given limits, as the area under a graph of that function between the given limits, provided that due regard is paid to signs (areas of regions below the horizontal axis must be treated as negative quantities). [M5.1, M5.2, P2.4]

can be evaluated: according to the fundamental theorem of calculus using

where is any indefinite integral of (i.e. any function that satisfies ). [M5.1, M5.2, P2.4]

also can be evaluated: by means of numerical integration. [M5.1, M5.2, P2.4]