average height of point on semicircle with respect to arc length

JNYANESHWAR

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i recently watched a lecture video where average height of a point on a semi circle with respect to arc length was found to be 2/pi. i dont understand what that means. i know what average height with respect to x means. average height with respect to x can be graphically interpreted as average height of rectangles of equal base dx under the curve. but what does average height with respect to arc length means graphically?? plz somebody clear my doubts...
 
Hears the way I look at it:
Average height wrt something means 'the normalizing factor' (what we divide by) is the 'total' of that something, in this case the total length of the arc of the semi circle, i.e. half of the circumference = π\displaystyle \pi r where r is the radius.

So we 'add up all the heights' and divide by π\displaystyle \pi r. In this case "add up all the the heights" means the integral of the heights. As was possibly shown in the video, to do this is to integrate (add up) in polar co-ordinates, i.e. the height is r sin(θ)\displaystyle (\theta) and the integral is from θ\displaystyle \theta =0 to θ=π\displaystyle \theta = \pi which gives the 'total of all the heights' as 2 r. Therefore the average is given by
A = 2rπr=2π\displaystyle \frac{2 r}{\pi r} = \frac{2}{\pi}
 
Hears the way I look at it:
Average height wrt something means 'the normalizing factor' (what we divide by) is the 'total' of that something, in this case the total length of the arc of the semi circle, i.e. half of the circumference = π\displaystyle \pi r where r is the radius.

So we 'add up all the heights' and divide by π\displaystyle \pi r. In this case "add up all the the heights" means the integral of the heights. As was possibly shown in the video, to do this is to integrate (add up) in polar co-ordinates, i.e. the height is r sin(θ)\displaystyle (\theta) and the integral is from θ\displaystyle \theta =0 to θ=π\displaystyle \theta = \pi which gives the 'total of all the heights' as 2 r. Therefore the average is given by
A = 2rπr=2π\displaystyle \frac{2 r}{\pi r} = \frac{2}{\pi}


thanks, but why does sum of heights varies?? with respect to x it is pi and with respect to arc length it is 2r.. how does changing co ordinates change sum of heights?? how is height with respect to arc length different from height with respect to x axis?? it may be silly question but plz help.
 
thanks, but why does sum of heights varies?? with respect to x it is pi and with respect to arc length it is 2r.. how does changing co ordinates change sum of heights?? how is height with respect to arc length different from height with respect to x axis?? it may be silly question but plz help.

Well, first of all I forgot an r in the other equation - I kept going back and forth between a semi-circle or unit radius and a radius of r. If you think about it, the average height has to depend on the radius and the radius is not in the other formula. It should have been "which gives the 'total of all the heights' as 2 r2. Therefore the average is given by
A = 2r2πr = 2rπ\displaystyle A\space =\space \frac{2r^2}{\pi r}\space =\space \frac{2r}{\pi}

The integral as well as the normalization is wrt the 'with respect to' variable. This changes our 'add up the heights' integral example to the integral of r sin(θ\displaystyle \theta) dθ\displaystyle \theta [a mix of radius of 1 and radius of r] to r sin(θ\displaystyle \theta) ds, where s is arc length. But since ds = r dθ\displaystyle \theta, we have an integrand of r2sin(θ\displaystyle \theta) dθ\displaystyle \theta and the rest follows.

For a 'with respect to x', the height is y so the integrand is y dx and the normalization is 2r which gives an average of
A = πr4\displaystyle \frac{\pi r}{4}
 
It is not the sum of heights that is varying! It is their relationship to the thing they are being compared to that is varying. The question was about "height of a point on a semi-circle" with respect to its arc length". That means "the height compared to the arc length" or, same thing, "the height divided by the arc length". If you were to find "height with respect to x" you would divide the height by the x value. It is not the height or sum of heights that is changing but the thing you are dividing by.
 
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