concept
in the interval for a given non-negative function of the , we want to determine the represented by curve < section>axis of the graph area of the clip, we can count an
Riemann integral core idea is to try to determine the integral value by approaching infinity . Also note that, as negative value, the value of the corresponding area of the also takes a negative value.
defined
1. divided sections
a closed interval [a, b] is a division P is a point to take a finite interval in this column . Each closed interval is called a sub-interval. Maximum is defined of subintervals of length: , where .
redefinition sample divided . Dividing one sampling a closed interval [a, b] after performing the dividing means , in each sub-interval extraction point . as defined above.
Fine segmentation : provided and form a closed interval [a, b] is a sample division, and is further divided. If for any , there such , and the presence of such , then put segmentation: , called segmentation , of a refined segmentation. Briefly, that after a segmentation is based on a division of the front to add some points and markers.
So we can define a partial order in all the samples in this interval is divided, called "Fine." If a segmentation is a further divided fine division, the former is more said "fine" than the latter.
2. Riemann and
to a closed interval [a, b] with a real-valued function defined , sampling and dividing Riemann definition of , of the following formula:
in the formula is a child interval length of the product of the function value at. Intuitively, the point is to the tag the X-axis distance is high, divided in subranges long rectangular area.
(a digital Lie Liman and upper right corner of the rectangle represents the sum of the area. This Lie Liman and tends to a constant value, denoted by Riemann integral of this function.)
3. Riemann integral
is not strictly speaking, the Riemann integral is divided when more and more "sophisticated" when, Riemann and trends limits. Here's proof, will "more and more 'sophisticated' 'make a strict definition.
is such that "more 'fine'" effective, require the tends to zero. Thus The function value only and close, and the difference between the area "under the curve" rectangular area will be smaller. In fact, this is probably describe Riemann integral defined.
strictly defined as follows :
is a function of on the closed interval [a, b] the Riemann integral, if and only if for any , there , so that the sampling for any segmentation , , a subinterval as long as its maximum length of , there:
that is, for a function , If on the closed interval [a, b], no matter how sampling is divided, as long as it is sufficiently small sub-interval the maximum length, the function are the Riemann sum tends to a certain value, then closed interval [a, b] in the presence of Riemann integral, and is defined as the Riemann sum limit, at this time, said function of Integrability a.
The definition of defects is not operational, because you want to test all sample split is difficult to do. Following the introduction of another definition, and then prove that they are equivalent.
another definition:
is a function of in the closed interval [a, b] the Riemann integral, if and only if for any , there is a sample division , , any such "fine" than their segmentation and , are:
these two definitions are equivalent. If there is a which satisfies the definition of a, then it also satisfies another. First, if there is a satisfying the first definition, then only need to take any one of the sub-interval the maximum length of segmentation. For finer than the division thereof, will clearly subinterval is less than the maximum length of the , thus satisfying
Riemann integral is generally defined as Darboux integral (i.e., the second definition), as integral Darboux integral Bi Liman simpler and more practical.
properties
1. Linear
Riemann integral is a linear transform, that is, if the and in the interval [a, b] the Riemann integrable, and is a constant, then:
Since a Riemann integral function is a real number, thus a fixed interval [a, b], the function provided a Riemann integrable Riemann integral to its mapping are all Riemannian a linear function of the product of the functional space.
2. Qualitative positive
(Lebesgue measure on sense) If the function is almost everywhere in the interval [a, b] is greater than or equal to 0, then it is [a, b] integral is also greater than zero on. If the in the interval [a, b on almost everywhere greater than or equal to 0, and it is integrated over the is equal to 0, then the almost everywhere zero.
3. Additivity
If the function can be accumulated in the [a, c] and [c, b] the interval, the may be accumulated in the [a, b] the interval, and there
whether a , b , how the size relationship between the c , the above relationships are established.
4. Other properties
1) real function [a, b] is the Riemann integrable, if and only if it is bounded and continuous almost everywhere.
2) If the [a, b] is a real function on Riemannian integrable Lebesgue it is integrable.
3) If is [a, b] a uniform convergent sequence on which limit of , then:
4) If a real function is monotonic in the interval, then it is integrable Riemann, because the discontinuous point set is countable set.
promote
can be generalized Riemann integral function value belonging to the dimensional space of. Linear integral defined, in particular, since the complex vector space is a real number, you can also define complex function value points.
Riemann integral only defined on a bounded interval, extended to infinite intervals not convenient. Perhaps the simplest extension is defined by the limits of integration, that is, as improper integrals (improper integral) the same.
Unfortunately, this is not very appropriate. Translation invariance (if a function pan left or right, it should remain unchanged Riemann integral) lost. General requirements and independent of the existence of the integral integration sequence. Even if this meet, is still not what we want, because the Riemann integral and consistent limits no longer have interchangeability. For example, let in the , the other fields equal to zero. All , . But uniformly converges to 0, the is 0 points. Thus . Even if this is the correct value, you can see an important criterion for the limit and ordinary integral exchangeable for improper integrals NA. This limits the application of the Riemann integral.
A better approach is to abandon the use of the Riemann integral and Lebesgue integral. Although Lebesgue integral Riemann integral extension of this point does not look obvious, but difficult to prove each Riemann integrable functions are Lebesgue integrable, and when both are integral value is defined consistent.
in fact a straightforward extension of the Riemann integral is Henstock-Kurzweil integral. Another approach
Riemann integral extension is replaced Riemannian accumulation definition factor , roughly speaking, this gives another sense, the integration of the length of the pitch. This is the Riemann - Stieltjes integral method employed
