I learned in my calc 1 class that to calculate the arc length of a curve, we are to compute the integral of the function. For example, the integral of a function that describes the path of a thrown baseball would give the total distance traveled by the baseball (I hope i'm using the term arc length correctly to describe this). 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.