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## Homework Statement

Calculate the area of the surface ##x^2+y^2+z^2 = R^2 , z \ge h , 0 \le h \le R##

## Homework Equations

##A(S_D) = \iint_D |\mathbf r'_s \times \mathbf r'_t|dsdt##

where ##S_D## is the surface over ##D##.

## The Attempt at a Solution

We write the surface in parametric form using spherical coordinates

##\mathbf r(\theta ,\phi) = R(\sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta)##

which gives us

##\mathbf r'_\theta = R(\cos \theta \cos \phi , \cos \theta \sin \phi , -\sin \theta)##

and

##\mathbf r'_\phi = R(-\sin \theta \sin \phi , \sin \theta \cos \phi , 0 )##

so we end up with

##|\mathbf r'_\theta \times \mathbf r'_\phi | = R^2|\sin \theta | |(\sin \theta \cos \phi , \sin \theta , \sin \theta \sin \phi , \cos \theta ) | = R^2|\sin \theta |##

and the area

##A(S_D) = \iint _D R^2|\sin \theta | = \int _?^? \int _0^{2\pi } R^2|\sin \theta | d\phi d? = 2\pi \int _?^? R^2|\sin \theta | d? ##

So my problem is i don't know what I am supposed to integrate over. I suppose it should be ##\theta## but I'm not sure between what limits. Or even if the switch to spherical coordinates was a good idea at all.

The answer according to the book should be ##A(S_D) = 2\pi R(R-h)##

which makes me think i made another error somewhere else as well since I got an excess of ##R##.

Appreciate any help. Cheers!