In my mind, there are two approaches to finding the average height of a semicircle. Integrating Cartesianally:
Or integrating polar:
Both of these seem intuative as you are integrating the height of all possible inputs, then dividing by the range of inputs to give you the average height of all inputs. Yet, we get different results from each.
Which is the correct average height, and why are they different? Or are they correct in their own manner, and why?