-- Consider the solid that lies above the square (in the xy-plane) R = [0, 2] ti

Question

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Consider the solid that lies above the square (in the xy-plane) R = [0, 2] times [0, 2], and below the elliptic paraboloid z = 25 - x2 - Ay2. Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners. Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. What is the average of the two answers from (A) and (B)? Using iterated integrals, compute the exact value of the volume.

Explanation / Answer

To divide R into 4 equal squares, we divide the x and y axes into 2 equal parts.

?x = ?y = (2 - 0)/2

We obtain the "grid"

(0, 1),(0, 2)

(1,1),(2, 1)

Note that each square has area ?x ?y = 4/4 = 1

(a)Picking the lower left points of each small square yields an approximate volume

i.e. Volume = (1) [f(0,0) + f(1,0) + f(0,1) + f(1,1) ]

= [25 + 24 + 21 + 20] =90

(b)Picking the upper rights points of each small square yields an approximate volume

i.e. Volume = (1) [f(1,1) + f(2,1) + f(1,2) + f(2,2) ]

= [20 + 17+ 8 + 5] = 50

(c) avg volume = (90 +50)/2 =70

(d) integrate [ [integrate{(25x - x^2 -4*y^2)dx} ] dy] from x =0 to 2 and y =0 to 2,

volume obtained = 84

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