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:
It can be deduced that the volume of the wedge cut from the first octant will be 20.
From the information, the wedge cut from the first octant will be z ≥ 0 and y ≥ 0 = 12−3y² ≥ 0 = 0 ≤ y ≤ 2
Also, it can be deduced that 0 ≤ y ≤ 2 and x = 2-y ⟹ 0 ≤ x ≤ 2. Therefore, V = ∫dzdydx. In this case,
dz = 12−3y
dy = 2-x
dx = 2
V = ∫dzdydx = ∫(12−3y²) dydx = ∫12(2-x)-(2-x)³dx
= [24(2) - 6(2)² + (2-2)⁴/4 -(2-0)⁴]/4
= 20
In conclusion, the volume of the wedge cut from the first octant will be 20.
Learn more about volume on:
https://brainly.com/question/12410983
