Why can't $\phi$ and $\theta$ be interchanged when calculating bounds for a sphere?
The volume of the unit sphere can be calculated with the triple integral:
$$\iiint_R \,dV\,= \frac{4}{3}\pi$$
where $R$ is bounded by: $$0\leq r \leq 1$$ $$0\leq \phi \leq \pi$$ $$0\leq \theta \leq 2\pi$$
But if the values of $\phi$ and $\theta$ are interchanged, the integral yields an incorrect result of 0.
I believe the same region would be covered, is my intuition wrong?
The volume differential $dV$ has a factor of $\sin \phi$. Switching to $\sin \theta$ gives a different value.