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If $f$ is a positive function, the intuitive interpretation of the Riemann integral

$\int_a^b f(x) dx$

is the area under the curve $f$ between $a$ and $b$.

Suppose $f$ and $g$ are smooth positive functions. Is it correct to interpret the Riemann-Stieltjes integral

$\int_a^b f(x) d g(x)$ as the volume under a "ribbon", where the height of the ribbon at a point $u$ between $a$ and $b$ is determined by $f$ and the thickness is determined by $g'$?

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It can't be volume, $f(x)$ and $\mathrm{d}g(x)$ are supposed to have the same dimensions of "length" and area units are the square of length units. –  J. M. Sep 10 '10 at 12:10
    
That's not quite satisfying as an answer- we have a limiting sum of the form ∑f(xi)(g(xi+1)−g(xi))\sum f(x_i)(g(x_{i+1}) - g(x_i)), which is a sum of square length units. Even if this weren't the case, the integral as defined above is dimensionless. –  Simon Sep 10 '10 at 13:36
    
Sorry, latex derp. That should read $\sum f(x_i)(g(x_{i+1}) - g(x_i))$ –  Simon Sep 10 '10 at 13:42
    
@J.M.: The dimensional analysis works out because the thickness of the "ribbon" is $dg/dx$, which is dimensionless. –  Rahul Sep 10 '10 at 17:39
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1 Answer

You are effectively replacing $\int_a^b f(x) dg(x)$ with $\int_a^b f(x) g'(x) dx$. A quick trip to Wikipedia reveals that this is perfectly fine when $g$ is absolutely continuous, or when it has a continuous derivative, but may not be valid otherwise:

If $g$ should happen to be everywhere differentiable, then the integral may still be different from the Riemann integral $$\int_a^b f(x) g'(x) \, dx,$$ for example, if the derivative is unbounded. But if the derivative is continuous, they will be the same. This condition is also satisfied if $g$ is the (Lebesgue) integral of its derivative; in this case $g$ is said to be absolutely continuous.

However, $g$ may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, $g$ could be the Cantor function), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of $g$.

Edit: I didn't notice the word "smooth" in your question. In this light, the answer to your question is yes.

Also, MathWorld is probably a better reference: the two expressions are equal "if $f$ is continuous and $g'$ is Riemann integrable over the specified interval".

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