The question I'm trying to solve asks us to find $\iint_{S}\vec{r}\cdot \vec{dS}$ over the surface described by the paraboloid $z = a^2 - x^2 - y^2$. They offer the parameterisation that $x=a\sin(\theta)\cos(\phi), y=a\sin(\theta)\sin(\phi), z = a^2 \cos^2(\theta)$.
It then follows that we can find $\vec{dS}$ by doing $\frac{\partial \vec{r}}{\partial \phi} \times \frac{\partial \vec{r}}{\partial \theta} d\phi d\theta$, where $\vec{r} = a\sin(\theta)\cos(\phi) \hat{i} + a\sin(\theta)\sin(\phi)\hat{j} + a^2 \cos^2(\theta) \hat{k}$. We then carry out the dot product in cartesian coordinates.
However, I tried doing this in spherical coordinates, and something went wrong, and I'm not exactly sure what went wrong, so it will be amazing if you can let me know. Here is my working and reasoning:
Since we were given the parameterisation for the coordinates of x, y, and z that describes the surface of the paraboloid, I assumed we can find the position vector $\vec{r} = r(\theta, \phi)\hat{r}(\theta, \phi)$ by saying that $r(\theta, \phi) = x^2 + y^2 + z^2 = a\sqrt{\sin^2(\theta) + a^2\cos^4(\theta)}$. After this, I carried out $\frac{\partial \vec{r}}{\partial \phi} \times \frac{\partial \vec{r}}{\partial \theta} d\phi d\theta$ as normal, using spherical coordinates instead of cartesian like before. Similarly, I then did the dot product with $\vec{r}$ in spherical coordinates. However this gave me an integral that was very obviously not the answer.
I have a few ideas as to why my thing is wrong: 1) my expression of the position vector is wrong, I thought it would be right, but if I try to convert that to cartesian it doesn't check out. 2) I found $\vec{dS}$ wrong, and the equation I used only works if $\vec{r}$ is in cartesian coordinates.
I am a bit lost now, and would appreciate any help.