Calculate the Riemann Stieltjes integral

Question

This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class.

Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ Calculate the Riemann-Stieltjes integral $$\int_0^4x^2d\alpha(x)$$ from the definition.

I'm getting an answer of 170, but don't know if what I'm trying to do is correct. — Alyse, May 22 '13 at 19:00
What do you mean by $\alpha(t)=n^2$ ? Is $\alpha(t)$ a constant? — Graham Hesketh, May 22 '13 at 19:13
$\alpha$ is a step function, so I think I can just evaluate the integral around the jumps — Alyse, May 22 '13 at 19:14
http://math2.eku.edu/jones/analysis_e_084.pdf Theorem 86 should be helpful, and I got $58$, still need to check — TTY, May 22 '13 at 19:26

score 5 · Answer 1 · answered May 22 '13 at 19:21

5

The Riemann-Stieltjes integral $\int_a^b f(x) dg(x)$ is the limit as the step size tends to zero of $\sum_{i=0}^{n-1}f(c_i)(g(x_{i+1})-g(x_i))$ for $a=x_0<x_1<...<x_n=b$ where $c_i$ is between $x_i$ and $x_{i+1}$.

$\alpha(x_{i+1})-\alpha(x_i)$ is $0$ unless $x_i$ and $x_{i+1}$ straddle the boundary between integers, which in the limit happens at $1,2,3$.

$\int_0^4 x^2 \mathrm{d}\alpha(x)=1^2(1^2-0^2)+2^2(2^2-1^2)+3^2(3^2-2^2)=58$

answered May 22 '13 at 19:21

Angela Pretorius

10,280

Why do you not have to take 4 into account? This is how I worked the problem but included $4^2(4^2-3^2)$ – Alyse May 22 '13 at 19:27
The $x_i$ go up to $4$. To straddle $4$ some $x_i$ would have to go over it. – Angela Pretorius May 22 '13 at 19:28

Calculate the Riemann Stieltjes integral

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