Riemann-Stieltjes integral

[Colloden]

For a piece for course work I have to solve a bounded Riemann- Stieltjes integral which is in the form

Integral of (x^2)dF(x) where F(x) is defined for some values of x and the integral has bounds.

Does anyone know of a worked example for something of a similar type? All I can find are explanations, proofs and so on – I can’t seem to find any worked examples for Riemann-Stieltjes integrals and I’m not sure what I’m doing.
Does anyone know of some?

 

[Vorpal]

If you have a particular differentiable F, you can just do it via integration by parts:
[latex]\int x^2\;{\mathrm d}F = x^2F - 2\int F\;{\mathrm d}x[/latex]
On the other hand, if F is monotonic and continuous, then
[latex]\int_a^b f(x)\;{\mathrm d}F = \int_{F(a)}^{F(b)} f(F^{-1}(t))\;{\mathrm d}t[/latex]
whenever both exist, because the terms in the Riemann-Stieltjes sum over partition {x_k} of [a,b] correspond to partition of {t_k = F(x_k)} of [F(a),F(b)]:
[latex]f(\xi_k)(F(x_k) - F(x_{k-1})) = f(F^{-1}(\tau_k))(t_k - t_{k-1})[/latex]
where τ_k = F(ξ_k) is in [t_k-1,t_k] due to monotonicity.

So in that case, it's straightforward to turn a Riemann-Stieltjes integral into a regular Riemann integral, and trivially extended to cases where F has finitely many turning points. You should be able to do a Riemann integral.

 

[Vorpal]

Hmm there is typo of a missing x in the right-side integral on the first line.

 

[Colloden]

Thanks Vorpal, thats helpful. What I could do with is a worked example.