I suppose you mean the AREA.

Not really, in this particular case. You can get by with just a little geometry+trigonometry, since you can determine the area of that part of the ellipse by multiplying the corresponding area of the circle over its major axis (with diameter 8) by the factor 1/2 (which is the ratio between minor and major axis of the ellipse).but I have no idea how to do it... I assume it would require integration though.