Calculate $$ I=\iint_{x^2+y^2+z^2=1}{e^{x-y} \mathbb{d}S} $$
Parameterization is not helpful: $$ I=\int_0^{2\pi}{\mathbb{d}\varphi\int_0^\pi{e^{\sin\theta(\cos\varphi-\sin\varphi)}\sin\theta\mathbb{d}\theta}} $$ ... nor is transformation to standard double-integral: $$ I=\int_{-1}^1{\mathbb{d}x\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2}{\frac{e^{x-y}}{\sqrt{1-x^2-y^2}}\mathbb{d}y}} $$