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However, we also learned that finding the integral of a function gives you the area under the curve described by the function. I must have a fundamental misunderstanding of either one or both of these statements, because if they are accurate, then what happens when you take the integral of a semi-circle? Then multiply it by 2? Something doesn't work out because I know the length of the curve (circumference of a circle, C=2pi(r)) is different than the area under/in-between the curves (A=pi(r)^2).

I would greatly appreciate if someone could please help me understand this apparent paradox.