Spherical coordinate system

Question

I can easily write $z$ axis value is $r\cos\theta$ but what will be for $x$ and $y$ axis, explain a bit please.

From the above how can I write the area element as $d\vec{a} = r^2\sin\theta d\theta d\phi\hat{r}$?

Answer 1

The piece of the sphere with radius between $r$ and $\mathrm{d}r$ etc. has a shape tends towards a cuboid as the size of the piece cut decreases, with height $\mathrm{d}r$, width $\mathrm{d}\theta$ and length $r\sin\theta\mathrm{d}\phi$. Its volume is $r^2\sin\theta \mathrm{d}\theta \ \mathrm{d}\phi\mathrm{d}r$.