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An indicator function of a bounded set is Riemann-integrable if and only if the set is Jordan measurable. The Riemann integral can be interpreted measure-theoretically as the integral with respect to the Jordan measure.
A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.
The region that is bounded can be seen as the area inside and . For example, the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is defined on the interval [ 2 , 4 ] {\displaystyle [2,4]} ∫ 2 4 x 3 d x {\displaystyle \int _{2}^{4}x^{3}\,dx} with the limits of integration being 2 {\displaystyle 2} and 4 {\displaystyle 4} .
A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell). Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval.
The fence is the section of the g(x)-sheet (i.e., the g(x) curve extended along the f(x) axis) that is bounded between the g(x)-x plane and the f(x)-sheet. The Riemann-Stieltjes integral is the area of the projection of this fence onto the f(x)-g(x) plane — in effect, its "shadow". The slope of g(x) weights the area of the projection. The ...
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. [5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval.
If the function space of locally integrable functions, i.e. functions belonging to (), is considered in the preceding definitions 1.2, 2.1 and 2.2 instead of the one of globally integrable functions, then the function space defined is that of functions of locally bounded variation.
Sometimes integrals may have two singularities where they are improper. Consider, for example, the function 1/((x + 1) √ x) integrated from 0 to ∞ (shown right). At the lower bound of the integration domain, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly ...