Evaluating Riemann-Stieltjes Integrals
We will now look at evaluating some Riemann-Stieltjes integrals. Before we do, be sure to recall the results summarized below. Let $f$ and $g$ be Riemann-Stieltjes integrable function with respect to $\alpha$ and $\beta$ on the interval $[a, b]$ and let $c \in \mathbb{R}$.
- Additivity of the Integrand: $\displaystyle{\int_a^b [f(x) + g(x)] \: d \alpha (x) = \int_a^b f(x) \: d \alpha (x) + \int_a^b g(x) \: d \alpha (x)}$.
- Homogeneity of the Integrand: $\displaystyle{\int_a^b cf(x) \: d \alpha (x) = c \int_a^b f(x) \: d \alpha (x)}$.
- Additivity of the Integrator: $\displaystyle{\int_a^b f(x) \: d [\alpha (x) + \beta(x)] = \int_a^b f(x) \: d \alpha (x) + \int_a^b f(x) \: d \beta (x)}$.
- Homogeneity of the Integrator: $\displaystyle{\int_a^b f(x) \: d[c \alpha(x)] = c \int_a^b f(x) \: d \alpha (x)}$.
- Integration by Parts: $\displaystyle{\int_a^b f(x) \: d \alpha (x) + \int_a^b \alpha (x) \: d f(x) = f(b)\alpha(b) - f(a)\alpha(a)}$.
- Reduction to a Riemann-Integral: If $f$ is bounded on $[a, b]$, $\alpha'$ exists and is continuous on $[a, b]$ then $\displaystyle{\int_a^b f(x) \: d \alpha(x) = \int_a^b f(x) \alpha'(x) \: dx}$
Let's now look at some examples.
Example 1
Evaluate the Riemann-Stieltjes integral $\int_0^1 x \: d x^2$.
We see that the integrand $f(x) = x$ is bounded on $[0, 1]$, the derivative of the integrator $\alpha (x) = x^2$ exists and is $\alpha'(x) = 2x$ and is continuous on $[0, 1]$, so we reduce the Riemann-Stieltjes integral above to get:
(1)Example 2
Evaluate the Riemann-Stieltjes integral $\int_0^{\pi} x \: d \cos x$.
Using integration by parts gives us that:
(2)Example 3
Evaluate the Riemann-Stieltjes integral $\int_0^{\pi} (x + 1) \: d (\sin x + \cos x)$.
Using the additivity of the integrand we have that:
(3)Using the additivity of the integrator and we have that:
(4)The Riemann-Stieltjes integral $\int_0^{\pi} x \: d \cos x = -\pi$ from example 1. The other three integrals can be evaluated by using Integration by parts:
(5)Therefore:
(8)