Darboux integral Guide, Meaning , Facts, Information and Description

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In real analysis, a branch of mathematics, the Darboux integral 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. Darboux integrals have the advantage of being simpler to define than Riemann integrals. Darboux integrals are named after their discoverer: Gaston Darboux.

Table of contents

1 Definition 2 Facts about the Darboux integral 3 See also 4 External link

Definition

A partition of an interval [a, b] is a finite sequence a = x₀ < x₁ < x₂ < ... < x_n = b. Each [x_i, x_i+1] is called a subinterval of the partition. A refinement of the partition x₀, ..., x_n is a partition y₀, ..., y_m such that for every i with 0≤i≤n, there is an integer r(i) such that x_i=y_r(i). In other words, to make a refinement, one cuts the subintervals into smaller pieces and does not remove any cuts.

Let f:[a,b]→R be a function, and let x₀, ..., x_n be a partition of [a, b]. Let:

The upper Darboux sum of f with respect to x₀, ..., x_n is

The lower Darboux sum of f with respect to x₀, ..., x_n is

The upper Darboux integral of f is

The lower Darboux integral of f is

If U_f = L_f, then we say that f is Darboux-integrable and set ∫f to be the common value of the upper and lower Darboux integrals.

Facts about the Darboux integral

If y₀, ..., y_m is a refinement of x₀, ..., x_n, then and

If x₀, ..., x_n and y₀, ..., y_m are two partitions (one need not be a refinement of the other), then . It follows that L_f ≤ U_f.

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if x₀, ..., x_n and t₀,...,t_m-1 together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of f corresponding to x_n and t₀,...,t_m-1 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.

See also

• Riemann integral
• Lebesgue integration

External link

• Article on the Darboux integral by the Science Fair Project Encyclopedia

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