For example, consider this question:
Write a triple integral, including limits of integration, that gives the specified volume: Under the sphere $^2+^2+^2= 9$ and above the region between $=$ and $= 2− 2$ in the $$-plane in the first quadrant.
I realize that this can be solved with just one integral (the middle integral would be with respect to $x$ and the outer with respect to $y$).
But I'm wondering whether this is also a valid solution:
$$\int_0^1 \int_0^x \int_0^{\sqrt{9-x^2-y^2}}dzdydx + \int_1^2 \int_{2x-2}^x \int_0^{\sqrt{9-x^2-y^2}}dzdydx $$
I've seen this done with double integrals, but I'm not sure whether it works nicely with triple integrals or whether some extra considerations have to be made...