Look at the above image. If we split up the ellipse thingy into two and put them in the regions colored blue and red, then, we get the following image -
As you can see now, there is a right-isosceles triangle in the middle of the quarter-circle. It has a base and height of , so its area is .
The quarter-circle has an area of
A small GIF showing all the stuff I said above, if it was to hard to understand -
Thus, the orange area is
An alternative solution for those who think the above is felony (lol)
As proved in the preface, the area of the regions shaded blue in the above image have an area of each.
Now, the remaining white regions are the regions shaded red in the image below. These have radii of length each, so they're areas are each.
So, the total white area is . Now, the orange area is equal to the area of the main quarter-circle minus the white regions area.
We have already calculated that the area of the main to be .
Thus, again the answer is