Why do the lengths of the dθ segments change, while the lengths of dx do not. I can see that the x-axis is a straight line, and the half-circle is curved, but I thought the segments of dθ would be the same length -- like dividing a line or function into infinitesimally small segments for integrating.
My integral is ∫(sinθ)dθ [0, pi]. I expected the segments of dθ to be equal. Intuitively, I thought that the average height (y value) with respect to arc length would be less than the continuous average. It is. And, intuitively, I thought the average would be weighed down by the near vertical sides of the half-circle.
The only connection I have made to the reason why the length of the segments of dθ change are because the extend a horizontal length of the length of the segment of dx. But my integral is with respect to dθ. Following that reasoning, why would the segments of dθ correspond to lengths of dx.
I understand that the average with respect to arc length is smaller because of the 1/b-a factor for an average.
