I hope this is a useful question, since I did not find any equivalent thread.
Assume you have an integral that depends on one limit of integration such as
$\int_a^b \phi(\tau, b) d\tau$
Now, I would like to perform a change of coordinates such as $\tau = f(\sigma)$, where $\sigma$ is the new integration variable. Assume that there is a relation between $d\tau$ and $d\sigma$: $d\tau = f' \; d\sigma=v \;d\sigma$. My question is the limit of integration $b$ in the $\phi$ function has to transformed or not in the new variable coordinates. In other words, which one of the following expression is right?
$\int_{f^-1(a)}^{f^-1(b)} \phi(f(\sigma), b) v d\sigma$
$\int_{f^-1(a)}^{f^-1(b)} \phi(f(\sigma), f^-1(b)) v d\sigma$
I personally think that the first expression is the right one, but I want to be sure about that.
Thank you for any reply and suggestion. Best regards,
Neostek
Well, $b$ is just a constant here, not a variable of integration or anything like that, so your first expression is right. An easy way to see this is replacing $b$ by a number, say $1$, then if we make the substitution $\tau=f(\sigma)$, we would simply have $$\int_a^1 \phi(\tau,1)\ d\tau=\int_{f^{-1}(a)}^{f^{-1}(1)} \phi(f(\sigma),1)f'(\sigma)\ d\sigma$$ Also, be careful with using $f^{-1}$, since both $f^{-1}(a)$ and $f^{-1}(b)$ may be the same value, and then you have to separate the integral into parts when $\sigma$ is increasing and decreasing.