Derivation of Riemann Stieltjes integral with floor function

calculus definite-integrals integration

Related problems: (I). Here is a theorem you can apply it to the problem,

Theorem: Suppose $f$ and $g$ are bounded functions with no common discontinuities on the interval $[a,b]$, and the Riemann-Stieltjes integral of $f$ with respect to $g$ exists. Then the Riemann-Stieltjes integral of $g$ with respect to $f$ exists, and $$\int_{a}^{b} g(x)df(x) = f(b)g(b)-f(a)g(a)-\int_{a}^{b} f(x)dg(x)\,. $$

Note that,

$$\int_{3}^{6}[x]dx = \int_{3}^{4} 3 dx + \int_{4}^{5} 4 dx + \int_{5}^{6} 5 dx \,. $$

Now, what do you think the value of the following integral is?

$$ \int_{3}^{6}d[x] = ?$$

Just apply the above theorem and see what you get.

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