what is the area of this Neumann Oval ? We calculate the area of the Neumann oval, which is the image of the unit disk in R2 under the mapping z + z^2/2. For this we need a new way of changing variables which involves the derivative squared. This method uses Jacobians, double integrals, polar coordinates. At the end we explain why there is a |f’z|^2 term, but it basically follows from the Cauchy Riemann equations. This is a must see for complex analysis and multivariable calculus lovers!