What is the geometrical meaning of a Riemann-Stieltjes integral?

Question

Geometrical meaning of Reimann integration is to find the area under the curve of function and $x$-axis. Due to this geometrical interpretation, all theorems on Riemann integration are easily understandable. I want to know what is the geometrical meaning of Riemann-Stieltjes integration. How can one see its graphical representation? Is it similar to finding the area between the curve and $x$-axis, or is it something else? Please someone explain with an example (without mathematical solution) with geometrical meaning.

Answer 1

Since, Stieltjes integral is the generalization of Riemann integral so, basic ideas on R integral is must before getting into the R-S integral or simply Stieltjes integral.

Ok, i would like to give the geometrical interpretation of Steiltjes integral so that you will get the intuitive meaning of R-S integral.

GEOMETRICAL INTERPRETATION OF R-S INTEGRAL:

Here in R-S integral, let’s consider we are integrating the function $f(\xi )$ with respect to the monotonic function $\alpha (\xi )$within the interval$[a,b]$.

mathematically we represent as below,

$$\int_{a}^{b}f(\xi )d\alpha (\xi )$$

let’s get into the geometrical interpretation of above integral.

we generally take the area of the $f(\xi )$with respect to the x-axis within certain interval in Reimann integral. Similarly we are calculating the area but a bit complex than the Reimann integral.

let’s take three axes where we keep $\xi $ and the functions $f(\xi )$ and $\alpha (\xi )$ at x-axis,y-axis and z-axis respectively. Now, we erect the wall from the curve traced by those functions then we can have the junction curve(since the intersection of the planes is curve). let’s take a light source emitting the parallel rays of light along the x-axis which results in the formation of shadow behind the wall on $f− \alpha$ plane as shown in image description,

Hence, the R-S integral gives the area of the shadow.