Kinetic energy of body rotating at constant angular speed

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Homework Statement



A straight wire has uniform density and total mass M. The wire is bent to form a closed loop, one section of which is a semi-circle of radius a, and the other section the diameter joining the two ends of the semicircle. The body is free to move about the midpoint O of its straight edge. It rotates at constant angular speed about an axis through O in the plane of the body and perpendicular to its straight edge.

Using T=Jijij/2 or otherwise, show that T=(1/12)Ma^2ω^2((3π+4)/(π+2))


Homework Equations



Jij=∫ρ(rkrkδij-rirj)dV

The Attempt at a Solution



I've worked out that ρ=M/(a(π+2))

I'm struggling with how to do the integrals, as its such an odd shape off to the examples we've done (cubes, etc). I tried to do the semi-circle bit and the diameter bit separately, but then I don't get aything like the answer I'm supposed to. Any pointers would be appreciated! :)
 

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