this only makes sense if you are limiting it to the first octant.
in that case we can set #z = 0# and note that the plane and the x-y plane meet along the line #5x + 5y = 6# [which can also be written #x = 6/5 - y# or #y = 6/5 - x#]
The plane in the first octant is shaded yellow and intercepts shown
The volume we can get most simply from a double integral
#int_{y = 0}^{6/5} int_{x = 0}^{6/5 - y} dx dy qquad ( z(x,y) )#
#= int_{y = 0}^{6/5} int_{x = 0}^{6/5 - y} dx dy qquad (6 - 5x - 5y)#
#=int_{y = 0}^{6/5} dy qquad [ 6x - (5x^2)/2 - 5xy]_{x = 0}^{6/5 - y} #
#=int_{y = 0}^{6/5} dy qquad ( 6(6/5 - y) - (5(6/5 - y)^2)/2 - 5(6/5 - y)y ) #
#=1/10 int_{y = 0}^{6/5} dy qquad ( (6 - 5y)^2 ) #
# = 1/10 [- 1/15 (6 - 5y)^3 ]_{y = 0}^{6/5} #
# = 1/150 [ (6 - 5y)^3 ]_{y = 6/5}^{0} #
#= 36/25#
