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Why do we use an integral for the heavier chain in this problem?

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Why do we use an integral for the heavier chain in this problem?
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You're right, it's not getting any lighter. It's getting heavier. At least as far as the lighter cable is concerned. The start even at the bottom, so you're not actually lifting any of the heavier chain until you get started, and more of it is being pulled up past the bottom/middle point.
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imagine the heavier chain was on the ground. you use a motor to lift the chain. the motor lifts the first link, now in order to lift the second link it has to also lift the first link. so the motor has to work harder and harder the more chain it draws up.
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