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Could someone check my work please? The question said to write an integral in spherical coordinates and evaluate it to find the mass of the hemisphere: $x^2+y^2+z^2\leq4; z\geq0$ if the density $\rho(x,y,z)$ is equal to $2z$.

My work:

$m= \iiint_Q\rho(x,y,z) dV $

$m$ = $\displaystyle \int_0^\frac{\pi}{2}\int_0^{2\pi}\int_0^22\rho^3\cos\phi\sin\phi \ d\rho \ d\theta \ d\phi = 8\pi$

Gabriel
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  • Is it possible that you mixed up the density ($\rho$) with the radius? – Thomas Jan 02 '15 at 03:05
  • Assuming you did the integral right, it looks correct. – IAmNoOne Jan 02 '15 at 03:10
  • $\rho^2\sin(\phi)$ comes from the transform into spherical coordinates and $2\rho\cos(\phi)$ comes form the transformation of $\rho$ = density = $2z$ into spherical (sorry for the confusion -- I used $\rho$ to denote both density and radius) – Gabriel Jan 02 '15 at 03:10
  • I see, and yes, that's correct. – Thomas Jan 02 '15 at 03:25

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