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Let me try to elaborate the question by an example:

I want to calculate the length of a straight line like $y=x$ between $[0, 1]$.

By conventional way I would do a definite integration of integrand $\sqrt{dx^2 + dy^2}$.

But why I can not choose other kind of "measure" of that "infinitesimal part"? For example, an integrand looks like Manhattan distance $dx + dy$, or probability?

Is there any criterion to choose which kind of integrand in integration?

P.S. You may notice that I use the word "measure". Actually I do not know the exact definition of that word in mathematics.

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You cannot use just any infinitesimal quantity; that will change the value of the integral. Thinking of the integral as a limit of finite approximations, rather than a mysterious thing involving infinitely many infinitesimals, should make it clearer what is happening. The following derivation should be enlightening:… – Samuel May 23 '13 at 6:04
Since you mention "manhattan" disctance, the question in my view becomes then: What do you define as "distance"? The integrand you mention works for our regular distance, when you look at for example the Poincare distance, that integrand looks slightly different, because that's how it is in that world... – imranfat May 23 '13 at 15:46 seems that I did not state my question clearly. Yes, I understand I need to choose different integrand based on different definition of "distance". But further I want to know why a particular integrand(is that so called measure?) can be integrated to get correct length/area/volume. In low dimensions(<=3) and with real life appreciable measure(length/area/volume), the intuition can help me to determine the correct integrand. But in higher dimension or with an "artificial" measure(like probability) I find it very difficult. – craftsman.don May 24 '13 at 2:13

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