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"Let's find the moment of inertia around this end."
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So, what you should imagine in that case is--
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you stick a nail through here
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and the guy rotates like that on a big circle
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with that as the center.
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If you stuck the nail through this midpoint,
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it's a different story,
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and the answer is different.
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So here,
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what I do is I take a distance x.
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I take a sliver of thickness dx.
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That sliver is small enough