In a book I've found the following, very simple, change of variables:

$$\int_{V} dV=\int_{M}\frac{dM}{\rho},$$

$$\text{with}\quad \rho dV=dM ,\quad \rho=\text{constant}$$

Can someone please explain me why the integration domain became simple $M$ and not M/$\rho$ ?


  • 1
    $\begingroup$ In this case I would guess $M$ is something like "the set all masses" so $M/\rho$ doesn't really make sense. $\endgroup$ – By Symmetry Aug 31 '16 at 10:22
  • $\begingroup$ So it is like saying "moving our attention" from the volume to the mass? Because the integration domain is not specifically defined (like from V(1) to V(2))? $\endgroup$ – horowitz Aug 31 '16 at 10:26
  • $\begingroup$ essentially yes. The integration domain is perfectly well defined in both cases, it is simply that it is not being defined in the form of an interval between 2 real numbers, so there is nothing to divide. You have to work out how to express the old domain in the new variables some other way. $\endgroup$ – By Symmetry Aug 31 '16 at 10:32
  • $\begingroup$ There's an assumption here that the density is constant. $\endgroup$ – garyp Aug 31 '16 at 10:44

When we take the integral with respect to the volume of the body, we are writing an integral over the body's volume $V$, and hence our bounds are in terms of the volume of the body. Thus, when we transform our integral to an integration with respect to the body's mass, we integrate over the mass of the body $M$, not its $M/\rho$, which would be the same as its volume. In each case, we are integrating to find a volume, and on the left-hand side, we integrate the volume directly, while in the second we integrate over the mass, indicating that the bounds of integration are going to be related to the mass of the body, as opposed to the volume of the body.

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