In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.[1] The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.[2] Moreover, the definition is readily extended to defining Riemann–Stieltjes integration.[3] Darboux integrals are named after their inventor, Gaston Darboux (1842–1917).
The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any boundedreal-valued function on the interval The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition. These ideas are made precise below:
Darboux sums
Lower (green) and upper (green plus lavender) Darboux sums for four subintervals
Each interval is called a subinterval of the partition. Let be a bounded function, and let
be a partition of . Let
The upper Darboux sum of with respect to is
The lower Darboux sum of with respect to is
The lower and upper Darboux sums are often called the lower and upper sums.
Darboux integrals
The upper Darboux integral of f is
The lower Darboux integral of f is
In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively.
and like Darboux sums they are sometimes simply called the lower and upper integrals.
If Uf = Lf, then we call the common value the Darboux integral.[4] We also say that f is Darboux-integrable or simply integrable and set
An equivalent and sometimes useful criterion for the integrability of f is to show that for every ε > 0 there exists a partition Pε of [a, b] such that[5]
Properties
For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (b−a) and height inf(f) taken over [a, b]. Likewise, the upper sum is bounded above by the rectangle of width (b−a) and height sup(f).
The lower and upper Darboux integrals satisfy
Given any c in (a, b)
The lower and upper Darboux integrals are not necessarily linear. Suppose that g:[a, b] → R is also a bounded function, then the upper and lower integrals satisfy the following inequalities.
For a constant c ≥ 0 we have
For a constant c ≤ 0 we have
Consider the function:
then F is Lipschitz continuous. An identical result holds if F is defined using an upper Darboux integral.
Examples
A Darboux-integrable function
Suppose we want to show that the function is Darboux-integrable on the interval and determine its value. To do this we partition into equally sized subintervals each of length . We denote a partition of equally sized subintervals as .
Now since is strictly increasing on , the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point. The starting point of the -th subinterval in is and the end point is . Thus the lower Darboux sum on a partition is given by
similarly, the upper Darboux sum is given by
Since
Thus for given any , we have that any partition with satisfies
which shows that is Darboux integrable. To find the value of the integral note that
Since the rational and irrational numbers are both dense subsets of , it follows that takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition we have
from which we can see that the lower and upper Darboux integrals are unequal.
Refinement of a partition and relation to Riemann integration
When passing to a refinement, the lower sum increases and the upper sum decreases.
A refinement of the partition is a partition such that for all i = 0, …, n there is an integerr(i) such that
In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.
If is a refinement of then
and
If P1, P2 are two partitions of the same interval (one need not be a refinement of the other), then
and it follows that
Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if and together make a tagged partition
(as in the definition of the Riemann integral), and if the Riemann sum of is equal to R corresponding to P and T, then
From the previous fact, Riemann integrals are at least as strong as Darboux integrals: if the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. There is (see below) a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.
Details of finding the tags
For this proof, we shall use superscripts to index and variables related to it.
Let be a sequence of arbitrary partitions of such that , whose tags are to be determined.
By the definition of infimum, for any , we can always find a such that Thus,
Let , we have
Taking limits of both sides,
Similarly, (with a different sequences of tags)
Thus, we have
which means that the Darboux integral exist and equals .
darboux, integral, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, february. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Darboux integral news newspapers books scholar JSTOR February 2013 Learn how and when to remove this template message In the branch of mathematics known as real analysis the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function Darboux integrals are equivalent to Riemann integrals meaning that a function is Darboux integrable if and only if it is Riemann integrable and the values of the two integrals if they exist are equal 1 The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral Consequently introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral rather than the true Riemann integral 2 Moreover the definition is readily extended to defining Riemann Stieltjes integration 3 Darboux integrals are named after their inventor Gaston Darboux 1842 1917 Contents 1 Definition 1 1 Darboux sums 1 2 Darboux integrals 2 Properties 3 Examples 3 1 A Darboux integrable function 3 2 A nonintegrable function 4 Refinement of a partition and relation to Riemann integration 5 See also 6 Notes 7 ReferencesDefinition EditThe definition of the Darboux integral considers upper and lower Darboux integrals which exist for any bounded real valued function f displaystyle f on the interval a b displaystyle a b The Darboux integral exists if and only if the upper and lower integrals are equal The upper and lower integrals are in turn the infimum and supremum respectively of upper and lower Darboux sums which over and underestimate respectively the area under the curve In particular for a given partition of the interval of integration the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum respectively of f in each subinterval of the partition These ideas are made precise below Darboux sums Edit Lower green and upper green plus lavender Darboux sums for four subintervals A partition of an interval a b displaystyle a b is a finite sequence of values xi such that a x 0 lt x 1 lt lt x n b displaystyle a x 0 lt x 1 lt cdots lt x n b Each interval x i 1 x i displaystyle x i 1 x i is called a subinterval of the partition Let f a b R displaystyle f a b to mathbb R be a bounded function and let P x 0 x n displaystyle P x 0 ldots x n be a partition of a b displaystyle a b Let M i sup x x i 1 x i f x m i inf x x i 1 x i f x displaystyle begin aligned M i sup x in x i 1 x i f x m i inf x in x i 1 x i f x end aligned The upper Darboux sum of f displaystyle f with respect to P displaystyle P is U f P i 1 n x i x i 1 M i displaystyle U f P sum i 1 n x i x i 1 M i The lower Darboux sum of f displaystyle f with respect to P displaystyle P is L f P i 1 n x i x i 1 m i displaystyle L f P sum i 1 n x i x i 1 m i The lower and upper Darboux sums are often called the lower and upper sums Darboux integrals Edit The upper Darboux integral of f is U f inf U f P P is a partition of a b displaystyle U f inf U f P colon P text is a partition of a b The lower Darboux integral of f is L f sup L f P P is a partition of a b displaystyle L f sup L f P colon P text is a partition of a b In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively L f a b f x d x U f a b f x d x displaystyle begin aligned L f equiv underline int a b f x mathrm d x 6pt U f equiv overline int a b f x mathrm d x end aligned and like Darboux sums they are sometimes simply called the lower and upper integrals If Uf Lf then we call the common value the Darboux integral 4 We also say that f is Darboux integrable or simply integrable and set a b f t d t U f L f displaystyle int a b f t dt U f L f An equivalent and sometimes useful criterion for the integrability of f is to show that for every e gt 0 there exists a partition Pe of a b such that 5 U f P ϵ L f P ϵ lt e displaystyle U f P epsilon L f P epsilon lt varepsilon Properties EditFor any given partition the upper Darboux sum is always greater than or equal to the lower Darboux sum Furthermore the lower Darboux sum is bounded below by the rectangle of width b a and height inf f taken over a b Likewise the upper sum is bounded above by the rectangle of width b a and height sup f b a inf x a b f x L f P U f P b a sup x a b f x displaystyle b a inf x in a b f x leq L f P leq U f P leq b a sup x in a b f x The lower and upper Darboux integrals satisfy a b f x d x a b f x d x displaystyle underline int a b f x dx leq overline int a b f x dx Given any c in a b a b f x d x a c f x d x c b f x d x a b f x d x a c f x d x c b f x d x displaystyle begin aligned underline int a b f x dx amp underline int a c f x dx underline int c b f x dx 6pt overline int a b f x dx amp overline int a c f x dx overline int c b f x dx end aligned The lower and upper Darboux integrals are not necessarily linear Suppose that g a b R is also a bounded function then the upper and lower integrals satisfy the following inequalities a b f x d x a b g x d x a b f x g x d x a b f x d x a b g x d x a b f x g x d x displaystyle begin aligned underline int a b f x dx underline int a b g x dx amp leq underline int a b f x g x dx 6pt overline int a b f x dx overline int a b g x dx amp geq overline int a b f x g x dx end aligned For a constant c 0 we have a b c f x c a b f x a b c f x c a b f x displaystyle begin aligned underline int a b cf x amp c underline int a b f x 6pt overline int a b cf x amp c overline int a b f x end aligned For a constant c 0 we have a b c f x c a b f x a b c f x c a b f x displaystyle begin aligned underline int a b cf x amp c overline int a b f x 6pt overline int a b cf x amp c underline int a b f x end aligned Consider the function F a b R F x a x f t d t displaystyle begin aligned amp F a b to mathbb R amp F x underline int a x f t dt end aligned then F is Lipschitz continuous An identical result holds if F is defined using an upper Darboux integral Examples EditA Darboux integrable function Edit Suppose we want to show that the function f x x displaystyle f x x is Darboux integrable on the interval 0 1 displaystyle 0 1 and determine its value To do this we partition 0 1 displaystyle 0 1 into n displaystyle n equally sized subintervals each of length 1 n displaystyle 1 n We denote a partition of n displaystyle n equally sized subintervals as P n displaystyle P n Now since f x x displaystyle f x x is strictly increasing on 0 1 displaystyle 0 1 the infimum on any particular subinterval is given by its starting point Likewise the supremum on any particular subinterval is given by its end point The starting point of the k displaystyle k th subinterval in P n displaystyle P n is k 1 n displaystyle k 1 n and the end point is k n displaystyle k n Thus the lower Darboux sum on a partition P n displaystyle P n is given by L f P n k 1 n f x k 1 x k x k 1 k 1 n k 1 n 1 n 1 n 2 k 1 n k 1 1 n 2 n 1 n 2 displaystyle begin aligned L f P n amp sum k 1 n f x k 1 x k x k 1 amp sum k 1 n frac k 1 n cdot frac 1 n amp frac 1 n 2 sum k 1 n k 1 amp frac 1 n 2 left frac n 1 n 2 right end aligned similarly the upper Darboux sum is given by U f P n k 1 n f x k x k x k 1 k 1 n k n 1 n 1 n 2 k 1 n k 1 n 2 n 1 n 2 displaystyle begin aligned U f P n amp sum k 1 n f x k x k x k 1 amp sum k 1 n frac k n cdot frac 1 n amp frac 1 n 2 sum k 1 n k amp frac 1 n 2 left frac n 1 n 2 right end aligned Since U f P n L f P n 1 n displaystyle U f P n L f P n frac 1 n Thus for given any e gt 0 displaystyle varepsilon gt 0 we have that any partition P n displaystyle P n with n gt 1 e displaystyle n gt frac 1 varepsilon satisfies U f P n L f P n lt e displaystyle U f P n L f P n lt varepsilon which shows that f displaystyle f is Darboux integrable To find the value of the integral note that 0 1 f x d x lim n U f P n lim n L f P n 1 2 displaystyle int 0 1 f x dx lim n to infty U f P n lim n to infty L f P n frac 1 2 Darboux sums Darboux upper sums of the function y x2 Darboux lower sums of the function y x2 A nonintegrable function Edit Suppose we have the Dirichlet function f 0 1 R displaystyle f 0 1 to mathbb R defined as f x 0 if x is rational 1 if x is irrational displaystyle begin aligned f x amp begin cases 0 amp text if x text is rational 1 amp text if x text is irrational end cases end aligned Since the rational and irrational numbers are both dense subsets of R displaystyle mathbb R it follows that f displaystyle f takes on the value of 0 and 1 on every subinterval of any partition Thus for any partition P displaystyle P we have L f P k 1 n x k x k 1 inf x x k 1 x k f 0 U f P k 1 n x k x k 1 sup x x k 1 x k f 1 displaystyle begin aligned L f P amp sum k 1 n x k x k 1 inf x in x k 1 x k f 0 U f P amp sum k 1 n x k x k 1 sup x in x k 1 x k f 1 end aligned from which we can see that the lower and upper Darboux integrals are unequal Refinement of a partition and relation to Riemann integration Edit When passing to a refinement the lower sum increases and the upper sum decreases A refinement of the partition x 0 x n displaystyle x 0 ldots x n is a partition y 0 y m displaystyle y 0 ldots y m such that for all i 0 n there is an integer r i such that x i y r i displaystyle x i y r i In other words to make a refinement cut the subintervals into smaller pieces and do not remove any existing cuts If P y 0 y m displaystyle P y 0 ldots y m is a refinement of P x 0 x n displaystyle P x 0 ldots x n then U f P U f P displaystyle U f P geq U f P and L f P L f P displaystyle L f P leq L f P If P1 P2 are two partitions of the same interval one need not be a refinement of the other then L f P 1 U f P 2 displaystyle L f P 1 leq U f P 2 and it follows that L f U f displaystyle L f leq U f Riemann sums always lie between the corresponding lower and upper Darboux sums Formally if P x 0 x n displaystyle P x 0 ldots x n and T t 1 t n displaystyle T t 1 ldots t n together make a tagged partition x 0 t 1 x 1 x n 1 t n x n displaystyle x 0 leq t 1 leq x 1 leq cdots leq x n 1 leq t n leq x n as in the definition of the Riemann integral and if the Riemann sum of f displaystyle f is equal to R corresponding to P and T then L f P R U f P displaystyle L f P leq R leq U f P From the previous fact Riemann integrals are at least as strong as Darboux integrals if the Darboux integral exists then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral so any Riemann sum over the same partition will also be close to the value of the integral There is see below a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral and consequently if the Riemann integral exists then the Darboux integral must exist as well Details of finding the tagsFor this proof we shall use superscripts to index P n displaystyle left P n right and variables related to it Let P n displaystyle left P n right be a sequence of arbitrary partitions of a b displaystyle a b such that P n 0 displaystyle P n to 0 whose tags are to be determined By the definition of infimum for any ϵ gt 0 displaystyle epsilon gt 0 we can always find a t i n x i n x i 1 n displaystyle t i n in left x i n x i 1 n right such that inf x x i n x i 1 n f x f t i n ϵ displaystyle inf x in left x i n x i 1 n right f x geq f t i n epsilon Thus i 0 N n 1 f t i x i 1 n x i n i 0 N n 1 inf x x i n x i 1 n f x ϵ x i 1 n x i n i 0 N n 1 inf x x i n x i 1 n f x x i 1 n x i n i 0 N 1 ϵ x i 1 n x i n i 0 N n 1 inf x x i n x i 1 n f x x i 1 n x i n ϵ b a displaystyle begin aligned sum i 0 N n 1 f t i x i 1 n x i n amp leq amp sum i 0 N n 1 left inf x in left x i n x i 1 n right f x epsilon right x i 1 n x i n amp amp amp sum i 0 N n 1 inf x in left x i n x i 1 n right f x x i 1 n x i n sum i 0 N 1 epsilon x i 1 n x i n amp amp sum i 0 N n 1 inf x in left x i n x i 1 n right f x x i 1 n x i n epsilon b a end aligned Let ϵ 1 n b a displaystyle epsilon 1 n b a we have i 0 N n 1 f t i x i 1 n x i n i 0 N n 1 inf x x i n x i 1 n f x x i 1 n x i n 1 n L f P n 1 n displaystyle begin aligned sum i 0 N n 1 f t i x i 1 n x i n amp leq amp sum i 0 N n 1 inf x in left x i n x i 1 n right f x x i 1 n x i n frac 1 n amp amp L f P n frac 1 n end aligned Taking limits of both sides R f i 0 N n 1 f t i x i 1 n x i n lim n L f P n lim n 1 n lim n L f P n displaystyle begin aligned R f sum i 0 N n 1 f t i x i 1 n x i n leq lim n to infty L f P n lim n to infty frac 1 n lim n to infty L f P n end aligned Similarly with a different sequences of tags R f lim n U f P n displaystyle begin aligned R f geq lim n to infty U f P n end aligned Thus we have R f lim n L f P n lim n U f P n R f displaystyle R f leq lim n to infty L f P n leq lim n to infty U f P n leq R f which means that the Darboux integral exist and equals R f displaystyle R f See also EditRegulated integral Lebesgue integration Minimum bounding rectangleNotes Edit David J Foulis Mustafa A Munem 1989 After Calculus Analysis Dellen Publishing Company p 396 ISBN 978 0 02 339130 9 Spivak M 1994 Calculus 3rd edition Houston TX Publish Or Perish Inc pp 253 255 ISBN 0 914098 89 6 Rudin W 1976 Principles of Mathematical Analysis 3rd edition New York McGraw Hill pp 120 122 ISBN 007054235X Wolfram MathWorld Spivak 2008 chapter 13 References Edit Darboux Integral Wolfram MathWorld Retrieved 2013 01 08 Darboux integral at Encyclopaedia of Mathematics Darboux sum Encyclopedia of Mathematics EMS Press 2001 1994 Spivak Michael 2008 Calculus 4 ed Publish or Perish ISBN 978 0914098911 Equivalence of Darboux and Riemann integral dead YouTube link Retrieved from https en wikipedia org w index php title Darboux integral amp oldid 1129856433, wikipedia, wiki, book, books, library,