![]() Which is again what we expect from the standard table for a circular segment. And we will integrate $x$ from 0 to $r_0\cos(\alpha/2)$. For a a given position along the $x$ axis, the limits of $y$ range from $0$ to $x\tan(\alpha/2)$. The difficulty is just in getting the correct limits of the double integral. Where $\rho$ is the mass density per unit area, which looks simple enough. I'm going to use Mathematica to do the brute force algebra and integration). ![]() Since this is clearly a homework problem, I'm going to skip the algebra steps and just show you the core parts of the problem (i.e. Now, only looking at the top half we can break the piece up into two sections: $I_1$ is on the left and is a triangle and $I_2$ is on the right and is a right triangle with a circular hypotenuse. To start with, we will recognize that the symmetry about the $x$ axis lets us only work on the top half and then multiply by a factor of 2 in the end. The differential element dA has width dx and height dy, so dA dx dy dy dx. ![]() ![]() Calculating the moment of inertia about the $x$ axis is a fair deal more complicated than calculating it about the $z$ axis as in my other answer. To find the moment of inertia, divide the area into square differential elements dA at (x, y) where x and y can range over the entire rectangle and then evaluate the integral using double integration. Since you actually asked for the moment about the $x$ axis. The unit of dimension of the second moment of area is length to fourth power, L4, and should not be confused with the mass moment of inertia.
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