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**1. Homework Statement**

Consider a half disk (of uniform density) with the flat end lying on the x-axis, symmetric about the y-axis (i.e. being cut into two quarters by the y-axis). Calculate the moments of inertia about each of the axes.

**2. Homework Equations**

$$I_{rr}=\sum_{i}m_ir_i^2$$

**3. The Attempt at a Solution**

I just need some making sure that I'm setting up the problem correctly.

The distance to the x-axis from any point on the disk is ##y##. Or, alternatively, ##r\sin{\phi}.## So we find that, $$I_{xx}=\int_{0}^{R}\int_{0}^{\pi}(r^2\sin^2{\phi})\rho\,rdrd\phi,$$ where ##\rho## is the mass density per unit area. However, my instructor has in his notes that, $$I_{xx}=\int_{0}^{R}\int_{0}^{\pi}(r^2\cos^2{\phi})\rho\,rdrd\phi.$$ I'm not sure why that would be, as the distance from any point to the axis of rotation (x in the case of ##I_{xx}## is ##y=r\sin{\rho}.##