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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.2010-09-10
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    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.2010-09-10
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    Sorry, latex derp. That should read $\sum f(x_i)(g(x_{i+1}) - g(x_i))$2010-09-10
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    @J.M.: The dimensional analysis works out because the thickness of the "ribbon" is $dg/dx$, which is dimensionless.2010-09-10

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