Answer
$$0$$
Work Step by Step
$$Area=A=\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \cos (x+y+z) \ dx \ dy \ dz \\=\int_{0}^{\pi} \sin (\pi+y+z) dz - \int_{0}^{\pi} \sin (y+z) \ dy \ dz \\=\int_{0}^{\pi} [-\cos (\pi+y+z) ]_0^{\pi} \ dz + \int_{0}^{\pi} \cos (y+z)]_0^{\pi} \ dz \\=\int_{0}^{\pi} [-\cos (2 \pi+z) + 2 \cos (\pi+z)+\cos z] \ dz\\=\int_{0}^{\pi} [-\cos (z) + 2 \cos (z)+\cos z] \ dz \\=(-2) \int_{0}^{\pi} \cos z \ dz \\=-2 [\sin z]_0^{\pi} \\=-2[ \sin \pi -\sin 0] \\=0$$
