Question

(1 point) Consider the solid that lies above the square (in the xyxy-plane) R=[0,1]×[0,1]R=[0,1]×[0,1], and below the elliptic paraboloid z=36−x2+2xy−4y2z=36−x2+2xy−4y2. Estimate the volume by dividing RR into 9 equal squares and choosing the sample points to lie in the midpoints of each square.

Answer 1

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