Answer:
Check attachment for better understanding of the points
Step-by-step explanation:
Given that,
R = [0,1] × [0,1]
Elliptic paraboloid Z = 36—x²+2xy—4y²
To divide R into 9 equal squares, we need to divide the x and y axes into 3 equal parts.
The area of each square is
A = ⅓×⅓ = 1/9 square units
Choosing the sample points at the centre of each square. Check attachment for coordinates of each centre.
For S1 : (1/6, 1/6)
Z = 36—x²+2xy—4y²
Z = 36—(1/6)²+2(1/6)(1/6)—4(1/6)²
Z = 35.92
For S2 : (1/2, 1/6)
Z = 36—x²+2xy—4y²
Z = 36—(1/2)²+2(1/2)(1/6)—4(1/6)²
Z = 35.81
For S3 : (5/6, 1/6)
Z = 36—x²+2xy—4y²
Z = 36—(5/6)²+2(5/6)(1/6)—4(1/6)²
Z = 35.47
For S4 : (1/6, 1/2)
Z = 36—x²+2xy—4y²
Z = 36—(1/6)²+2(1/6)(1/2)—4(1/2)²
Z = 35.14
For S5 : (1/2, 1/2)
Z = 36—x²+2xy—4y²
Z = 36—(1/2)²+2(1/2)(1/2)—4(1/2)²
Z = 35.25
For S6 : (5/6, 1/2)
Z = 36—x²+2xy—4y²
Z = 36—(5/6)²+2(5/6)(1/2)—4(1/2)²
Z = 35.14
For S7 : (1/6, 5/6)
Z = 36—x²+2xy—4y²
Z = 36—(1/6)²+2(1/6)(5/6)—4(5/6)²
Z = 33.47
For S8 : (1/2, 5/6)
Z = 36—x²+2xy—4y²
Z = 36—(1/2)²+2(1/2)(5/6)—4(5/6)²
Z = 33.81
For S9 : (5/6, 5/6)
Z = 36—x²+2xy—4y²
Z = 36—(5/6)²+2(5/6)(5/6)—4(5/6)²
Z = 33.92
Then, the volume is estimated under the given paranoid as
Volume = ∑ Zi × A where i- ranges from 1 to 9.
Volume=
Z1+Z2+Z3+Z4+Z5+Z6+Z7+Z8+Z9)×A
Volume = (35.92 + 35.81 + 35.47 + 35.14+ 35.25 + 35.14 + 33.47 + 33.81+ 33.92) × 1/9
Volume = 313.93 × 1/9
Volume. = 34.88 Cubic unit
