P_1=\int_{-\sqrt 2}^{\sqrt 2}dx\int_{-\sqrt 2}^{\sqrt 2}\frac{\sqrt{4+y^2}}2\,dy=\int_{-\sqrt 2}^{\sqrt 2}(\ln(2+\sqrt 3)+\sqrt 3)\,dx=2\sqrt 2(\ln(2+\sqrt 3)+\sqrt 3)