Surface area of a sphere limits If I am finding the surface area of a sphere in spherical coordinates my intergral would be like this:
$$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\phi d\theta =4\pi R^2$$
But if I do the following:
$$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\theta d\phi =0$$
What makes the answers different? In both cases I am integrating over the same total area! What does the last integral represent?

I think the confusion is in your integration limits, each integral symbol with its limits corresponds to one and only one variable! Geometrically what you want is:
$$
\int^{2\pi}_0\left( \int^{\pi}_0 R^2 \sin (\theta) d\theta\right) d\phi =2R^2\int^{2\pi}_0d\phi = 4\pi R^2
$$
The problem arises when you integrate $\sin\theta$ from $0$ to $2\pi$, but in spherical coordinates $\theta\in[0,\pi]$.
