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