Stieltjes Integral meaning. Can anybody give a geometrical interpretation of the Stieltjes integral:
$$\int_a^bf(\xi)\,d\alpha(\xi)$$
How would we calculate?
$$\int_a^b \xi^3\,d\alpha(\xi)$$
for example.

Since, Steiltjes integral is the generalization of Reimann integral so, basic ideas on R integral is must before getting into the R-S integral or simply Steiltjes 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. 
Hence, the R-S integral gives the area of the shadow.
For your another question(calculation), you can use the relation between the Reimann integral and Reimann-Steiltjes integral,
$$\int_{a}^{b}\xi ^{3}d\alpha (\xi ) = \int_{a}^{b}\xi ^{3}{\alpha }'(\xi )d\xi $$
thanks!
