Solve Real-Life Problems w/ Stieltjes Integral

[r4nd0m]

Can you give me a simple real-life problem, where you need to use Stieltjes integral and can you show how you proceed in solving this kind of problems?

 

[HallsofIvy]

I'm not certain what you consider "real life"! I suppose it wouldn't help for me to point out that in "real life" you have to take a calculus test.

The Stieljes integral differs from the ordinary Riemann integral in that, after we have divided the interval (a to b, say) into n intervals with endpoints x_i, x_i+1, instead of defining [itex]Delta x_i to be simply x_i+1- x_i, that is, the length of the interval, we define it to be $\alpha(x_{i+1})- \alpha(x_i)$ where $\alpha(x)$ can be any increasing function. Taking the "Riemann sums" as usual then and taking the limit as the number of intervals goes to infinity results in the Stieltjes integral $\int f(x)d\alpha$ rather than the Riemann integral $\int f(x)dx$.

Of course if $\alpha(x)$ happens to be differentiable then it is easy to see that
$$\int f(x)d\alpha= \int f(x)\alpha'(x)dx$$

One common application is this: let $\alpha(x)$ be the "step" function (f(x)= 0 for 0<= x< 1, f(x)= 1 for 1<= x< 2, etc.). Then the sum
$$\Sum_{n=0}^\k f(n)$$
can be written as the Stieltjes integral
[tex]\int_0^{n+1} f(x)d\alpha[/itex]
allowing one to combine the theory of sums with integrals.


In particular