$\delta$ in spherical coordinates: $\int_0^R\int_0^{2\pi}\int_0^{\pi}\delta(\theta-\pi/2)(r^2\sin(\theta)\,d\theta \,d\phi \,dr)$

Question

Suppose you have a disc of radius $R$, we can find its area in polar coordinates by: $$\int_0^R\int_0^{2\pi}(r\,d\phi \,dr)=\pi r^2$$

Naively, I also expect to be able to integrate in spherical coordinates the function $\delta(\theta-\pi/2)$, specifying that the disc lies in the $\theta=\pi/2$ plane. However, this gives a factor of $R^3$:

$$\int_0^R\int_0^{2\pi}\int_0^{\pi}\delta(\theta-\pi/2)(r^2\sin(\theta)\,d\theta\, d\phi \,dr)=\frac{2}{3}\pi R^3$$

I believe something is going wrong with using the $\delta$ function, as the factor of $rd\theta$ comes from the variation in $\theta$, which now is no longer varying. But in that case, how do we now deal with delta functions in spherical coordinates?

Answer 1

http://www.fen.bilkent.edu.tr/~ercelebi/mp03.pdf

http://www.physics.usu.edu/wheeler/em3600/notes05coordinatesdiracdelta.pdf

Explained in both about as well as anyone can. Hope this helps.

I could have tried and looked brilliant by just copying the main discussion there and taking full credit, but I don't do that. I'd rather give good honest references then a phony ripoff explanation.