Can I use $y=r\cos(\theta)$, $z=r\sin(\theta)$, and $x=x$ as cylindrical coordinates?

Question

I need to compute $\displaystyle\int_{-\sqrt{2}}^{\sqrt{2}}\displaystyle\int_{-\sqrt{2-z^2}}^{\sqrt{2-z^2}}\displaystyle\int_{0}^{1+y^2+z^2}z^3dxdydz~$ using cylindrical or spherical coordinates.

Because we have $1+y^2+z^2$ and $\pm\sqrt{2-z^2}$, it would be useful to use cylindrical coordinates. However, my question is: can I define $y=r\cos(\theta)$, $z=r\sin(\theta)$, and $x=x$ for this problem? I'm not sure if I chan change variables whenever I use these coordinates, that's my main problem.

Thanks in advance.

Answer 1

Yes, that works fine. All that is happening is $r$ is the distance between the origin and the projection of the point onto the $yz$ plane and $\theta$ is the angle in the $yz$ plane between said projection and the positive $y$ axis, which is a perfectly valid picture.