Calculate integral $$\int\int_S z^2(1+xy)dxdy+y^2(1+xz)dydz+x^2(1+yz)dzdx$$ where $S$ is the inside of surface $x^2+y^2=z^2,0\leq z\leq 1$.
Now I assume this is a surface integral because $x^2+y^2=z^2$ is a surface of a $3D$ figure.
I'm wondering if $$\int\int_S z^2(1+xy)dxdy+y^2(1+xz)dydz+x^2(1+yz)dzdx=\int\int_Sz^2(1+xy)dxdy+\int\int_Sy^2(1+xz)dydz+\int\int_Sx^2(1+yz)dzdx$$?
Can I just put $x=r\cos \varphi,y=r\sin \varphi,z=r$ then we have $dxdy=rdrd\varphi$ ($r$ comes from Jacobian) and $dydz=-r\cos\varphi drd\varphi$ and $dzdx=-r\sin\varphi drd\varphi$
So the integral is equal to $$\int_0^1\int_{-\pi}^\pi r^3(1+r^2\sin\varphi\cos\varphi)drd\varphi+\int_0^1\int_{-\pi}^\pi -r^3\sin^2\varphi\cos\varphi(1+r^2\cos\varphi)drd\varphi+\int_0^1\int_{-\pi}^\pi-r^3\cos^2\varphi\sin\varphi(1+r^2\sin\varphi)$$
But I'm unsure if I can do this, do I have to care about orientation, can I just split the integral as I did and can I put the substitution I did?
You can split it, but then integrals are not taken over $S$, but over its projections onto $XOY,YOZ,XOZ$ respectively.