Let Σ be the unit upper hemisphere $x^2 + y^2 + z^2 = 1$, $z ≥ 0$. Calculate Integral $z^4 dA$.
I know z would be $cos(phi)$, but why is $dA=sin(phi) d(theta) d(phi)$? is it being the derivative of z a coincidence?
2026-03-27 18:05:47.1774634747
Understanding spherical coordinates
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If you have a parameterization $r(\theta, \varphi)$ for your surface, then $dA$ is given by $$\Vert r_\theta \times r_\varphi\Vert \,d \theta d\varphi$$ Here, $r = \langle \cos \theta \sin \varphi, \sin \theta \sin \varphi, \cos \varphi \rangle $, and the above formula gives us $dA = \sin \varphi\, d\varphi d\theta$. The similarity between $dA$ and the derivative of $z$ is a coincidence, in that it does not hold for other situations.