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## Homework Statement

A solid B occupies the region of space above ##z=0## and between the spheres ##x^2 + y^2 + z^2 = 16## and ##x^2+y^2+(z-1^2) = 1##. The density of B is equal to the distance from its base, which is ##z = 0##. The mass of the solid B is ##\frac{188\pi}{3}##. Find the coordinates of the center of mass.

## Homework Equations

Center of mass in z = ##\frac{1}{m} \iiint \limits_E z *\rho(x,y,z) dV##

## The Attempt at a Solution

The center of mass in x and y is at coordinate 0 because of the symmetry of the domain, so there's only the z-value to calculate.

So my integral is ## \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{2\cos\phi}^{4} \rho \cos\phi * \rho \cos\phi * \rho^2\sin\phi d\rho d\phi d\theta##, then divided by the mass, but the answer sheet (which is a student's graded paper) has ##\rho\cos\phi * \rho^2\sin\phi## as the term to integrate, and therein lies my question: why are they only integrating z once? Don't we have z (the density) * z (the term already present in the center of mass equation) as the term to integrate?

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