at the lecture of Work, average value, probability(23th lecture) when the professor tried to find the average height of a unit circle( a portion above the x-axis) with y=sin(θ）, he said that the relative weight of lower portion of the upper semi-circle is heavier. But when it comes to y=(1-x)^(1/2)( I mean changing the variable), this does not happen( all relative weight is equal) Why? I cannot understand this point.
MIT 18.01 Single Variable Calculus (OCW)
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Would you give a time frame for the video?
On the basis of what you said, I'm a little confused. You have y=sin(theta), but in polar coordinates there is no y variable. If it's y=sin(x), that is not a unit circle, it's a sine wave.
Work, average value, probability
I don't know how to think the 2nd question
can you help me?
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I have reviewed that portion of the lecture. The professor gives a recap of his reasoning at 14:30 to 15:00 in the video. I would recommend watching that portion of it once or twice. If you still have questions about it, see if you can pin down exactly where in his explanation that you begin to have problems.
I'm not sure where that problem came from that you attached. I would suggest plugging in values and doing the integration.