Evaluate ∫∫∫s Z dxdydz, where S is the solid bounded by x + y + z = 2, x = 0, y = 0, and z = 0.
Is s: 0≤z≤2 0≤y≤(2-z) 0≤x≤(2-y) ? and then we take the integral.
Not quite. The plane,which is equivalent to x= 2- y- z, crosses the yz-plane, where x= 0, at the line y+ z= 2. And that line, which is equivalent to z= 2- y, crosses the z axis, where y= 0, at z= 2. To cover that figure, z goes from 0 to 2. For each z, y goes from 0 to 2- z. And for each y, z, x goes from 0 to 2- y- z.