Find the volume of the wedge cut from the first octant by the cylinder z=12-3y^2 and the plane x+y=2.

Answer:The wedge cut from the first octant ⟹ z ≥ 0 and y ≥ 0 ⟹ 12−3y^2 ≥ 0 ⟹ 0 ≤ y ≤ 2   0 ≤ y ≤ 2 and x = 2-y ⟹ 0 ≤ x ≤ 2   V = ∫∫∫ dzdydx   dz has changed from zero to 12−3y^2  dy has changed from zero to 2-x  dx has changed from zero to 2   V = ∫∫∫ dzdydx = ∫∫ (12−3y^2) dydx = ∫ 12(2-x)-(2-x)^3 dx =  24(2)-6(2)^2+(2-2)^4/4 -(2-0)^4/4 = 20 Step-by-step explanation: