I need to find the integral of gaussian curvature of a surface given by the graph over disk x^{2}+y^{2}<=2. I know that the integral of gaussian curvature= 2pi*(euler characteristic).

The Euler characterisic of the disk x^{2}+y^{2}<=1 is 1. So would the Euler characterisic of my disk: x^{2}+y^{2}<=2 also be 1?

thanks