What is the average value of the positive y-coordinates of an ellipse?

[IHave]

Homework Statement

Find the average value of the positive y-coordinates of the ellipse x^2/a^2 + y^2/b^2 = 1

I don't know where to begin, please help.

 

[Dr. Seafood]

"Average" means to sum the numbers in a set and divide by how many you added; i.e. ${\Sigma x_n} \over {n}$. Since the ellipse has a continuous nature, to "sum" the y-coordinates means to integrate over an interval. The "number of addends" here is the length of the interval, so we divide by that.

First write y in terms of x. Note that you have to take a square root to do this, but the question only asks for the positive y-coordinates, so consider only the principal branch of y. Integrate with respect to x over the "top half" of the ellipse -- the interval [-a, a]. Divide by the length of this interval, which is 2a. Essentially, carry out the following calculation to solve the problem:

$${{1} \over {2a}}{\int^a_{-a} \sqrt{b^2 (1 - {x^2 \over a^2})} \quad dx}$$