TITLE: Triple integral probably in spherical Coordinates: $ \iiint _{W} zy\, dz\,dy\,dx$
QUESTION [0 upvotes]: I’m trying to solve this integral:
$$ \iiint _{W} zy\, dz\,dy\,dx\,,$$
where $W$ is the volume inside $ x^{2} +y^{2} +z^{2} =1$ and $ z^{2} =x^{2} +y^{2}$.
my first attempt was to write this in spherical coordinates the following way :
$$ \int ^{2\pi }_{0} \sin\theta \ d\theta \int ^{1}_{0} \rho ^{4} \ d\rho \int ^{\frac{\pi }{2}}_{0} \sin^{2} \varphi .\cos\varphi \ d\varphi $$
but I have a feeling this is very wrong and I also get $0$ as the answer!
any help would be appreciated.

REPLY [2 votes]: $zy$ is an odd function in both $z$ and $y$ and has even symmetry about the $xy$ and $xz$ planes (i.e. $(x,y,z)\in W \iff (x,\pm y,\pm z) \in W$) thus the integral will be zero.
{"set_name": "stack_exchange", "score": 0, "question_id": 3737289}