Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse

We use Gronwall's area formula to find the area of some differents regions as circles, ellipses and lemniscates.We use Laurent and Taylor series expansions of conformal mapping from the exterior of the unit disk to either of these regions to compute the area of them.We close this work with the discussion of orthogonal Tchebychev polynomials of second kind on the ellipse and interpolation.