What is the difference between Riemann Stieltjes integral and Riemann integral?

If α is a differentiable function, then the Riemann-Stieltjes integral ∫f(x)dα is the same as the Riemann integral ∫f(x)dαdxdx. However, if α is not differentiable (and it does not even have to be continuous) the Riemann-Stieljes integral will exist while the Riemann integral does not.

What is the point of the Riemann Stieltjes integral?

It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.

Why are Riemann sums important?

Riemann Sums give us a systematic way to find the area of a curved surface when we know the mathematical function for that curve. They are named after the mathematician Bernhard Riemann (pronounced “ree-man”, since in German “ie” is pronounced “ee”).

Why is Riemann Sums useful?

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution.

How do you evaluate Riemann-Stieltjes integrals?

Then the Riemann–Stieltjes can be evaluated as where the integral on the right-hand side is the standard Riemann integral. Cavalieri’s principle can be used to calculate areas bounded by curves using Riemann–Stieltjes integrals. The integration strips of Riemann integration are replaced with strips that are non-rectangular in shape.

Why is the Riemann integral the simplest integral?

The Riemann integral is the simplest integral to define, and it allows one to integrate every continuous function as well as some not-too-badly discontinuous functions. There are, however, many other types of integrals, the most important of which is the Lebesgue integral.

What are some examples of Riemann integrability?

Examples of the Riemann integral Let us illustrate the definition of Riemann integrability with a number of examples. Example 1.4. Define f : [0,1] → Rby f(x) = (1/x if 0 < x ≤ 1, 0 if x = 0. Then Z 1 0 1 x dx isn’t defined as a Riemann integral becuase f is unbounded. In fact, if

What is the Stieltjes integral?

The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.