Proof with Darboux integral - question.

The integral mean value theorem states that if f is continuous on [a,b], then there exists a c in [a,b] such that ∫[a,b] f(x) dx = f(c)(b-a). Since f is bounded by M, we have that f(c) <= M for all c in [a,b]. Therefore, ∫[a,b] f(x) dx = f(c)(b-a) <= M(b-a). This shows that ∫[a,b] f(x) < M(b-a).In summary, we can prove that for a Riemann integrable function f(x) bounded by a real number M, the integral of f(x) over the interval [a,b] is always
  • #1
peripatein
880
0
Hello,

Homework Statement


I was asked to prove that for f(x), Riemann integrable and bounded from above by real number M:
∫[a,b] f(x) < M(b-a)


Homework Equations





The Attempt at a Solution


Since f(x) is Riemann integrable, ∫[a,b] f(x) must be equal to both the lower and upper Darboux sums. Therefore: ∫[a,b] f(x) = supf(x) [a,b]Ʃ[i=1,n]Δxi = supf(x) [a,b](b-a) <= M(b-a)
I am really not sure this is rigorous enough, or even how it ought to be approached. I'd appreciate some guidance please.
 
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  • #2
I think this can be seen as a consequence of the integral mean value theorem.
 
1.

What is the Darboux integral?

The Darboux integral, also known as the upper and lower Darboux sums, is a method of calculating the area under a curve. It is defined as the limit of sums of rectangles that approximate the area beneath the curve. It is named after French mathematician Gaston Darboux.

2.

How is the Darboux integral different from other types of integrals?

The Darboux integral is different from other types of integrals, such as the Riemann integral, because it uses a series of rectangles to approximate the area under the curve, rather than dividing the area into smaller and smaller rectangles.

3.

What is the significance of the Darboux integral in mathematics?

The Darboux integral is significant in mathematics because it allows for the calculation of the area under a curve, which is important in many fields such as physics, engineering, and economics. It also has applications in more advanced mathematical concepts, such as the Fundamental Theorem of Calculus.

4.

What are the conditions for a function to be Darboux integrable?

In order for a function to be Darboux integrable, it must be bounded on a closed interval. This means that there is a finite number that the function cannot exceed on that interval. Additionally, the function must have a finite number of discontinuities on that interval.

5.

How is the Darboux integral calculated?

The Darboux integral is calculated by dividing the interval into smaller subintervals and finding the area of rectangles that approximate the area under the curve on each subinterval. The Darboux integral is then found by taking the limit of these sums as the subintervals become smaller and smaller.

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