Sorry I do not have enough reputation to upload picture. My question was best described by the picture I drew. So this is the link of the picture I drew.
Maybe I missed the critical section of fundamental of calculus. I don't get intuitively why finding average power is different if you separate the region like 1,2 and calculating the average of two then adding, versus finding the whole area then dividing the whole length. What makes it different? Isn't it the same process? Why do I have to divide 2 again to my calculated average?
Picture at the bottom summarize what I mean.
Let's say a car travels 900 km in 9 hours, then (in traffic) it travels 10 km in one hour. The average speed is (900+10)/(9+1)=91 km/h. If you would use your approach, it would be (900/9+10/1)/2=55 km/h. The point is that the left region has greater weight in the average, since the area is larger