Denoting the unit circle, , by we define an ellipse to be the set of points
- Show that the points on satisfy the equation:
- Prove that the area of is measurable and that
- Proof. If is a point on then is a point on (since all points of are obtained by taking a point of and multiplying the -coordinate by and the -coordinate by ). By definition of , we must then have,
- Proof. From part (a) we know is the set of points such that . But, this implies,
Hence, the area of is the area enclosed from to by the graphs of
To show this region is measurable and has area we begin with this identity (from Apostol, 2.4 Exercise 17),
In the second to last line we have used the expansion/contraction of the interval of integration. Hence, we know the integral from to of exists and has value . Thus, is measurable and
In the last line, (x/a) ^ 2 instead of x/a