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

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