What does integrating a function wrt to another function intuitively mean?

Asked Jul 10 '17 at 03:09

Active Jul 10 '17 at 03:09

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I sometimes see stuff that is of the following notation:

$$ \int f(x) dG(x) $$

I reckon a specific example would be:

$$ \int x \operatorname{d \sin(x)} $$

How can one intuitively interpret this. Does this mean that we're compressing and stretching the x-axis by a sin function?

asked Jul 10 '17 at 03:09

user1172468

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see this MO answer. – Dando18 Jul 10 '17 at 03:13
@Dando18 many thanks -- will check it out. – user1172468 Jul 10 '17 at 03:23
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Riemann-Stieltjes integral – Robert Israel Jul 10 '17 at 03:39
what does actually $ \int x \operatorname{d \sin(x)}$ mean ? formally..
Isn't just substitution ? i.e., $\int x \cos(x) \operatorname{d x}$
– Red shoes Jul 10 '17 at 03:50
@Ashkan -- it mean integrate x w.r.t to sin(x) -- my understanding is that it is not a substitution – user1172468 Jul 10 '17 at 04:31
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My question was exactly that . what does integrating $x$ w.r.t $\sin$ mean? – Red shoes Jul 10 '17 at 04:33
@Ashkan that was my question too :-) -- I'm trying to precisely understand what the notation means intuitively. – user1172468 Jul 10 '17 at 04:35
I thought You already knew formal definition but want get some intuition about that definition. ! – Red shoes Jul 10 '17 at 04:38
@Ashkan -- ah I see -- I actually work in the reverse most of the time -- I like to get a good feel of the notation means intuitively and then try to relate that to the formal definition – user1172468 Jul 10 '17 at 04:41
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https://math.stackexchange.com/q/295383/219176 – Red shoes Jul 10 '17 at 06:02

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