Riemann Stieltjes Integral Calculator

The Riemann-Stieltjes integral is a generalization of the Riemann integral that allows integration with respect to a function rather than just the independent variable. This calculator helps you compute Riemann-Stieltjes integrals for given functions f(x) and g(x) over a specified interval.

What is a Riemann-Stieltjes Integral?

The Riemann-Stieltjes integral extends the concept of the Riemann integral by allowing integration with respect to a function g(x) rather than just dx. This is particularly useful in probability theory, economics, and other fields where cumulative functions are involved.

The integral is defined as the limit of Riemann-Stieltjes sums as the partition becomes finer. The standard Riemann integral is a special case where g(x) = x.

Formula and Calculation

The Riemann-Stieltjes integral of f(x) with respect to g(x) over [a, b] is defined as:

∫[a,b] f(x) dg(x) = lim_{n→∞} Σ_{i=1}^n f(x_i) [g(x_i) - g(x_{i-1})]

Where:

f(x) is the integrand function
g(x) is the integrator function
a and b are the integration limits
x_i are points in the partition of [a, b]

Our calculator approximates this integral by dividing the interval into subintervals and summing the products of function values and differences in g(x).

Worked Example

Let's compute ∫[0,1] x d(x²):

Divide [0,1] into n subintervals with points x_i = i/n
Compute the sum Σ_{i=1}^n x_i [g(x_i) - g(x_{i-1})]
Take the limit as n → ∞ to get the exact value

The exact value of this integral is 1/3. Our calculator will provide this approximation for any given functions and interval.

Applications

Riemann-Stieltjes integrals are used in:

Probability theory (probability measures)
Economics (consumer surplus)
Physics (distributions)
Mathematical finance (stochastic calculus)

FAQ

What's the difference between Riemann and Riemann-Stieltjes integrals?: The Riemann integral integrates with respect to dx, while the Riemann-Stieltjes integral integrates with respect to any function g(x).
When should I use this calculator?: Use this calculator when you need to compute integrals with respect to a function rather than just dx.
How accurate are the results?: The calculator provides an approximation of the integral. For exact values, you may need symbolic computation software.
Can I use this for probability measures?: Yes, the Riemann-Stieltjes integral is particularly useful for probability measures where g(x) represents a cumulative distribution function.