It appears that the domains over which the integrals are being performed are mixed up. The last equation should be written as follows:
$S = \int_{0}^{2\pi} \int_{0}^{\pi} \sin(\theta) d\theta d\phi = 4\pi$
It appears that the domains over which the integrals are being performed are mixed up. The last equation should be written as follows:
$S = \int_{0}^{2\pi} \int_{0}^{\pi} \sin(\theta) d\theta d\phi = 4\pi$