We know that $\int f(x)dx$ represents the area of a curve where $x$ is an independent variable varying over all real numbers.But what does $\int h(g(x)) dg(x)$ represent?I see people using change of variables formula in integration and that boils down to like $\int z dz=\frac{z^2}{2}+c$ where we are usingthe properties proved for ordinary integration where x was an independent varible but here $z$ is a function of $x$.Could you please shed some light on what does the integral $\int h(g(x)) dg(x)$ mean?
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There's a 3D geometric interpretation of that in this question , if that is what you are looking for : https://math.stackexchange.com/questions/2388158/what-is-the-geometrical-meaning-of-a-riemann-stieltjes-integral#:~:text=What%20is%20the%20geometrical%20meaning%20of%20a%20Riemann%2DStieltjes%20integral%3F,-riemann%2Dintegration&text=Geometrical%20meaning%20of%20Reimann%20integration,Riemann%20integration%20are%20easily%20understandable. – Sarvesh Ravichandran Iyer Dec 22 '20 at 11:50
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just call $g(x)$ , $t$ , then it is no brainer. – jimjim Dec 22 '20 at 11:51
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I don't understand,could you be a bit more elaborate jimjim – a_i_r Dec 22 '20 at 12:10
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Teresa Lisbon My question was regarding $h(g(x))$ with respect to $g(x)$ but the link you gave shows $f(x)$ with respect to $g(x)$. – a_i_r Dec 22 '20 at 12:14
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just write it like that, what do you get – jimjim Dec 22 '20 at 12:46