Consider the solid that lies above the square (in the xy-plane) R=[0,2] x [0,2]
ID: 2834347 • Letter: C
Question
Consider the solid that lies above the square (in the xy-plane) R=[0,2] x [0,2] , and below the elliptic paraboloid z=81-(x2)-(3y2)
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 volume.
Explanation / Answer
For (A) and (B), divide [0,2] on both the x and y-axes in half.
This yields four squares (draw a picture to see this).
A) The lower left hand corners of these squares are
(0,0), (1,0), (0,1), and (1,1).
So, V = (2-0)/2 * (2-0)/2 * {f(0,0) + f(1,0) + f(0,1) + f(1,1)} = 1(81 + 80 + 78 + 77) = 317
B) The upper right hand corners of these squares are
(1,1), (2,1), (1,2), and (2,2).
So, V = (2-0)/2 * (2-0)/2 * {f(1,1) + f(2,1) + f(1,2) + f(2,2)} = 1(77 + 74 + 68 + 65) = 284
C) Average = (1/2) * (317 + 284) = 300.5.
D) V = integral(x in [0,2], y in [0,2]) (81 - x^2 - 3y^2) dy dx
= integral(x in [0,2]) (81y - x^2 y - y^3) {for y in [0,2]} dx
= integral(x in [0,2]) (162 - 2x^2 - 8) dx
= integral(x in [0,2]) (154 - 2x^2) dx
= 154x - 2x^3/3 {for x in [0,2]}
= 308-16/3 = 908/3=302.67
I hope this helps!