- #1

- 114

- 0

## Homework Statement

Use spherical coordinates to find the moment of inertia about the z-axis of a solid of uniform density bounded by the hemisphere [tex]\rho=cos\varphi[/tex], [tex]\pi/4\leq\varphi\leq\pi/2[/tex], and the cone [tex]\varphi=4[/tex].

## Homework Equations

[tex]I_{z} = \int\int\int(x^{2}+y^{2})\rho(x, y, z) dV[/tex]

## The Attempt at a Solution

I tried to convert that equation to cylindrical coordinates and got this (k representing density because it's uniform)

[tex]I_{z} = k \int^{2\pi}_{0}\int^{\pi/2}_{\pi/4}\int^{1}_{0}\rho^2 sin^{2}\varphi*d\rho*d\varphi*d\theta[/tex]

Plugged that into my calculator and got:

[tex]\frac{k\pi(\pi+2)}{12}[/tex]

The book answer is:

[tex]\frac{k\pi}{192}[/tex]

What am I doing wrong?