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 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 &quot;Riemann sums&quot; 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.<br /> <br /> Of course if \alpha(x) happens to be differentiable then it is easy to see that<br /> \int f(x)d\alpha= \int f(x)\alpha&amp;#039;(x)dx<br /> <br /> One common application is this: let \alpha(x) be the &quot;step&quot; function (f(x)= 0 for 0&lt;= x&lt; 1, f(x)= 1 for 1&lt;= x&lt; 2, etc.). Then the sum <br /> \Sum_{n=0}^\k f(n)<br /> can be written as the Stieltjes integral<br /> \int_0^{n+1} f(x)d\alpha<br /> allowing one to combine the theory of sums with integrals.<br /> <br /> <br /> In particular