TITLE: If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?
QUESTION [26 upvotes]: I have $y$ is a function of $t$.
I have reached a situation here where I need to evaluate
$$\displaystyle \int_0^b{\frac{dy}{dt}dt}$$
Now clearly $y$ has dependence on $t$, otherwise $\displaystyle \frac{dy}{dt}$ should be 0.
So now I write
$$\displaystyle \int_{y(0)}^{y(b)}{dy} = y \rvert_{y(0)}^{y(b)} = y(b) - y(0)$$
I know that dt's don't cancel, but what do you call it, then?  Just a "change of variables"?  How do we justify where $dt$ went?

REPLY [2 votes]: Yes, it is a change of variables.  In $$ \int_0^b \frac{dy}{dt} \,dt ,$$ the independent variable is $t$ and everything else (namely $y$) is treated as a function of $t$.  But in $$ \int_{y(0)}^{y(b)} dy ,$$ the independent variable is $y$ and (at least once you've worked out what constants $y(0)$ and $y(b)$ are) everything else (namely nothing) is treated as a function of $y$.  So you changed variables from $t$ to $y$.
As for where the $dt$ went, I wonder why you want to know?  It is possible (despite what people may tell you) to think of differentials as honest-to-goodness things, so that you are dividing by them in derivatives and multiplying by them in integrals, and in that case the $dt$s certainly do cancel.  But if you don't want to do that, if you treat the $dt$ as merely part of the notation, then it didn't go anywhere.  When the variable was $t$, you wrote $dt$ at the end of the integral, and when the variable became $y$, you wrote $dy$; that's what you write in this notation.
Perhaps you should ask where the $dy/dt$ went.  Since you changed variables from $t$ to $y$, it was replaced with $dy/dy$, which is $1$.  And then there is a quirk of this notation that you don't bother to write the integrand when it is $1$.  (This quirk makes sense if you think of differentials as actual things that you multiply by in an integral; multiplying by $1$ doesn't do anything and so can be omitted.  But if you think of them as just part of the notation, then this is just a quirk.)