Estimate the intgral81 – x* - y dA, where R is the square D= [0,4]× [0,4]. Divide D into four equal squares and choose the samplepoint to be the upper right corners of each square.O A. 976O B. No correct AnswerO C. 488O D. 1952O E. 244

Step:-1 Given that V=∬D 81-x2-y2dA, D=0, 4×0, 4 Here, f(x)=81-x2-y2 As given choose upper right…

Answered: Estimate the intgral JI81 -x -y" JdA,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Estimate the intgral
81 – x* - y dA, where R is the square D= [0,4]× [0,4]. Divide D into four equal squares and choose the sample
point to be the upper right corners of each square.
O A. 976
O B. No correct Answer
O C. 488
O D. 1952
O E. 244

Expert Solution

Step 1

Step:-1

Given that $V = \iint_{D} (81 - x^{2} - y^{2}) d A, D = [0, 4] \times [0, 4]$

Here, $f (x) = 81 - x^{2} - y^{2}$

As given choose upper right corner, so we'll take point H for ( I ) , C for ( II ), D for ( IV ) and E for ( III ).

step:-2

$∆ A = 2 \times 2 (s i d e o f s q u a r e) ∆ A = 4 V = \iint_{D} (81 - x^{2} - y^{2}) d A V \approx ∆ A (f (H) + f (C) + f (D) + f (E)) V \approx ∆ A (f (2, 2) + f (4, 2) + f (4, 4) + f (2, 4)) V = ∆ A (f (2, 2) + f (4, 2) + f (4, 4) + f (2, 4)) V = 4 (73 + 61 + 49 + 61) = 976 V = 976$

Hence, option (a) is true.

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