Use the Green’s Theorem area formula given above to find the areas of the regions enclosed by the curves.

One arch of the cycloid x = t - sin t, y = 1 - cos t

If a simple closed curve C in the plane and the region R it encloses satisfy the hypotheses of Green’s Theorem, the area of R is given by

The reason is that by Equation (4), run backward,

Green's Theorem Area Formula Area of R - 1 2 x dy y dx