The area under the cycloid
A cycloid is the curve traced by a point on a circle as the circle rolls along a line. (For technical reasons, this circle is rolling on a vertical line.)
Roberval (1634) proved that the region between a sine curve and the cycloid has the same area as half the circle.
This demonstration uses Cavalieri's principle, that the area of a 2-dimensional object is the integral of the lengths of cross sections (sum of the areas of the slices). So the positions of the slices doesn't matter.
Drag the "Cav" slider to move the slices from the region between the (vertical) cycloid and a (vertical) sine curve to the circle.
Details:
Half the cycloid is inscribed in a rectangle whose dimension are the diameter of the circle and half the circumference of the circle. The sine curve cuts this rectangle in half. Therefore the area under half of the cycloid is the same as one and one half the area of the circle, and the area under one arch of the cycloid is 3 times the area of the circle.