I have a complicated integral which schematically looks like \begin{equation} I = \int_{-1}^1\mathop{dt}\int_0^{\infty}\mathop{dx}\int_0^{2\pi}\mathop{d\phi} f(\phi,x,t), \tag{1} \end{equation} where the function $f$ is too complicated to write down here.
In addition to the bounds given on the integral, there is a restriction on the domain of integration \begin{equation} \vert \cos{\phi} \vert \leq x\sqrt{1-t^2}, \tag{2} \label{eq2} \end{equation} which you could think of as being enforced by a step function times the integrand.
Say we integrate out $\phi$ first, successfully taking into account the restriction \ref{eq2} . What does Eq. \ref{eq2} become after doing that integral?
If this was just a triple integral in cartesian coordinates, you could just set the integrated-out variable ($\phi$ in this case) to zero to derive the new condition on the remaining integral, but I believe it might be different here because the coordinate system is more complicated, since $\phi$ is an angular variable.