(1 point) Consider the solid that lies above the square (in the xy-plane) R = [0
ID: 2891502 • Letter: #
Question
(1 point) Consider the solid that lies above the square (in the xy-plane) R = [0,2] × [0, 21 and below the elliptic paraboloid z 64- z2-4y2 (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners (B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. (C) What is the average of the two answers from (A) and (B)? (D) Using iterated integrals, compute the exact value of the volumeExplanation / Answer
Solution:
z = 64 - x^2 - 4y^2 , R = [0,2] x [0,2]
x = y = (2 - 0)/2 = 1.
Here's a grid with these 4 squares:
(0,2)....(1,2)....(2,2)
(0,1)....(1,1)....(2,1)
(0,0)....(1,0)....(2,0)
(A) Using lower left corners,
V (1 * 1) [ z(0, 0) + z(1, 0) + z(0, 1) + z(1, 1)]
z(0,0) = 64 - x^2 - 4y^2 = 64 - 0 - 0 = 64
z(1,0) = 64 - x^2 - 4y^2 = 64 - 1^2 - 0 = 63
z(0,1) = 64 - x^2 - 4y^2 = 64 - 0^2 - 4*1^2 = 60
z(1,1) = 64 - x^2 - 4y^2 = 64 - 1^2 - 4*1^2 = 59
V = [64 + 63 + 60 + 59] = 246.
(B) Using upper right corners,
V (1 * 1) [z(1, 1) + z(2, 1) + z(1, 2) + z(2, 2)]
z(1,1) = 64 - x^2 - 4y^2 = 64 - 1^2 - 4*1^2 = 59
z(2,1) = 64 - x^2 - 4y^2 = 64 - 2^2 - 4*1^2 = 56
z(1,2) = 64 - x^2 - 4y^2 = 64 - 1^2 - 4*2^2 = 47
z(2,2) = 64 - x^2 - 4y^2 = 64 - 2^2 - 4*2^2 = 44
V = [59 + 56 + 47 + 44] = 206.
(C) (1/2)(246 + 206) = 226.
(D) The iterated integral is z dA
....2.2
= (64 - x^2 - 4y^2) dx dy
...0.0
....2
= [64x - (x^3)/3 - 4xy^2]02 dy
...0
....2
= [(64*2 - (2^3)/3 - 4*2*y^2) - (0)] dy
...0
....2
= [128 - 8/3 - 8y^2] dy
...0
= [128y - (8/3)y - (8/3)y^3]02
= [(128*2 - (8/3)*2 - (8/3)*2^3) - 0]
= 256 - 16/3 - 64/3
= 688/3 = 229.33
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