Curvilinear Coordinates to Solve Integrals

I get the idea of curvilinear coordinates, but don't understand how it would make corresponding symmetrical integrals easier to solve. How would you evaluate an integral, for simplicity's sake say $$\iint_{S}\textbf{r}\cdot d\textbf{S},$$ where S is the surface of a sphere with radius R, centered at the origin using polar spherical coordinates?

Would you just convert everything into Cartesians? Wouldn't that just defeat the point of working with curvilinear coordinates?

Well,in this case,you certainly wouldn't. At least,you'd be crazy to do it given how simple the equation becomes in spherical coordinates! This is really a triple integral in $R^{3}$ unless the integral is taken over a level surface.
• When I try to solve it I get: $\iint_{S}r(\hat{\textbf{r}}\cdot \hat{\textbf{r}})r^2sin\theta d\theta d\phi = \int_{0}^{2\pi}d\phi \int_{0}^{\pi}d\theta r^3sin\theta = 4\pi r^3$ ??? – user167289 Feb 11 '15 at 0:12