How to change the limits of integration
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-\frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $\infty$, $s$ decreases from $0$ to $-\infty$.
$s=-r^{2}$ gives $ds=-2rdr$ so $dr =-\frac 1 {2r} ds$. Also, as $r$ increases from $0$ to $\infty$, $s$ decreases from $0$ to $-\infty$.