Question

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

Answer 1

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`