Multivariable Calculus 2.6.4 - Triple Integrals and Volume

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The Tutor Wizard uses triple integrals over general bounded solids in 3-space to compute volumes.
0:39 - example 1
7:55 - example 2
13:15 - example 3
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Trying #2 before seeing you solve, I changed to polar coordinates and let z take the place of x (i.e. z = rcostheta)

It worked out and I got the right answer. I'm wondering if, in general, this approach is okay? Or if it could cause issues letting z = rcostheta.

michaelkuby