التكامل شرح 1

* partition of an interval


In mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the form
a = x₀ < x₁ < x₂ < ... < x_n = b.


Such partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral.


The norm (or mesh) of the partition
x₀ < x₁ < x₂ < ... < x_n


is the length of the longest of these subintervals; it is
max{ |x_i − x_i−1| : i = 1, ..., n }.


As the mesh approaches zero, a Riemann sum based on the partition approaches the Riemann integral.


A tagged partition is a partition of an interval together with a finite sequence of numbers t₀, ..., t_n−1 subject to the conditions that for each i,
x_i t_i x_i+1.


In other words, it is a partition together with a distinguished
point of every subinterval. The mesh of a tagged partition is defined
the same as for an ordinary partition. We can define a partial order
on the set of all tagged partitions by saying that one tagged partition
is bigger than another if the bigger one is a refinement of the smaller
one.


Suppose that together with are a tagged partition of [a,b], and that together with are another tagged partition of [a,b]. We say that and together are a refinement of together with if for each integer i with , there is an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with .
Said more simply, a refinement of a tagged partition takes the starting
partition and adds more tags, but does not take any away.


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*Darboux integral



Definition




A partition of an interval [a,b] is a finite sequence of values x_i such that



Each interval [x_i−1,x_i] is called a subinterval of the partition. A refinement of the partition



is a partition



such that for every i with



there is an integer r(i) such that



In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.


Let ƒ:[a,b]→R be a bounded function, and let



be a partition of [a,b]. Let






Lower (green) and upper (green plus lavender) Darboux sums for four subintervals



The upper Darboux sum of ƒ with respect to P is



The lower Darboux sum of ƒ with respect to P is



The upper Darboux integral of ƒ is



The lower Darboux integral of ƒ is



If U_ƒ = L_ƒ, then we say that ƒ is Darboux-integrable and set



the common value of the upper and lower Darboux integrals.




Facts about the Darboux integral







When passing to a refinement, the lower sum increases and the upper sum decreases.



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
.


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 ƒ corresponding to P and T is R, 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. It is not
hard to see that there is 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.

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