Suggest an ellipse with semiaxes a and b (a>b) and circle of radius b, the center of circle lying on the extension of the major axis of the ellipse. Show that for every line parallel to major axis of the ellipse, the portion of that line inside the ellipse will be a/b times the portion inside the circle. Use this fact and Cavalieri's principle to compute the area of the ellipse.
Cavalieri's principe implies that figures in a plane lying between two parallel lines and such such that all sections parallel to those lines have the same length must have equal area.