How do you find the average value of the positive y-coordinates of the ellipse #x^2/a^2 + y^2/b^2 = 1#?

1 Answer
George C.
Oct 9, 2015

Divide the area of the upper half of the ellipse by the length of the #x# axis of the ellipse to find that the average positive #y# coordinate is:

#((pi ab) / 2) / (2a) = (pi b) / 4#

Explanation:

This ellipse passes through #(a, 0)#, #(0, b)#, #(-a, 0)# and #(0, -b)#

The area of the ellipse is #pi a b#

So the area of the upper half of the ellipse is #(pi a b)/2#

The base of the upper half of the ellipse is of length #2a#

So the average positive #y# value is #((pi a b) / 2)/(2a) = (pi b) / 4#

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