I_i=\int_0^{\sqrt{3-i}}\int_{x^2+i}^{x^2+i+1}\sqrt{i+1}\,dy\,dx+\int_{\sqrt{3-i}}^{\sqrt{4-i}}\int_{x^2+i}^4\sqrt{i+1}\,dy\,dx=\sqrt{i+1}\left(\sqrt{3-i}+\int_{\sqrt{3-i}}^{\sqrt{4-i}}(4-i-x^2)\right)\,dx