definite integral,
n. 1. (informally) the expression for the evaluation of the indefinite integral of a positive function between two limits of integration, representing the area between the graph of the given function and the x-axis between these values of x. If the given limits of integration are a and b, and the interval [a, b] is divided into n equal subintervals of width δ x, then, as shown,
the blue shaded region is the limit as n tends to infinity of the sum of the areas of the rectangles constructed on each subinterval with height f(x) for some x in that subinterval; these are known as elements of area, of which the red shaded rectangle is an example. This limit is written
and is evaluated as F(b) - F(a), where f(x) is the given function, x = a and x = b are the limits of integration, and F(x) is the indefinite integral ∫ f(x) dx. See also fundamental theorem of calculus.
2. the actual value of such an expression.
3. (more properly) a. the definite integral is said to exist in the sense of the Riemann integral, or otherwise, if the appropriate limit of Darboux sums exists. The definite integral of a continuous positive function between a and b then gives the area enclosed by the curve and axis between those bounds. For general continuous functions, the integral is the algebraic sum of the enclosed areas above and below the x-axis, where the latter has negative sign.
b. more generally, the definite integral of a function is said to exist if the Lebesgue integral of the product of the function with the characteristic function of that interval exists; that is, provided that the function can be approximated by measurable simple functions. If the function is continuous, then this gives the Riemann integral as the limit of Darboux sums.
and is evaluated as F(b) - F(a), where f(x) is the given function, x = a and x = b are the limits of integration, and F(x) is the indefinite integral ∫ f(x) dx. See also fundamental theorem of calculus.
2. the actual value of such an expression.
3. (more properly) a. the definite integral is said to exist in the sense of the Riemann integral, or otherwise, if the appropriate limit of Darboux sums exists. The definite integral of a continuous positive function between a and b then gives the area enclosed by the curve and axis between those bounds. For general continuous functions, the integral is the algebraic sum of the enclosed areas above and below the x-axis, where the latter has negative sign.
b. more generally, the definite integral of a function is said to exist if the Lebesgue integral of the product of the function with the characteristic function of that interval exists; that is, provided that the function can be approximated by measurable simple functions. If the function is continuous, then this gives the Riemann integral as the limit of Darboux sums.