In the equation above for the MI for a rod, why are we taking the limits from -l/2 to l/2?
And why doesn't the integral doesn't include the centre of mass?
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The limits run where they are because you're taking the moment about the center of the rod, which means that the center (x=0) of your coordinate system is the middle of the rod.
As for your second question, it doesn't make a ton of sense to me -- the integral runs through the center of mass, but moment of inertia and center of mass are two different concepts.