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I have three problems to prove based on this:

1] Find the vector area of a hemispherical bowl of radius R.

I solved a bit of this.

The area element for a hemisphere is given by [tex]d\vec a = R^2\sin \theta d\theta d\phi \hat r[/tex].

So,[tex]\vec a = R^2\int_{0}^{\frac{\pi}{2}}\sin \theta d\theta \int_{0}^{2\pi} d\phi \hat r[/tex]

How am I supposed to integrate over the unit vector [itex]\hat r[/itex]?

2] Show [tex]\vec a = 0[/tex] for any closed surface.

3] Show [tex] \vec a[/tex] is the same for all surfaces sharing the same boundary.