ECOPY2_LDAPMAIL_12092008-095656

ECOPY2_LDAPMAIL_12092008-095656 - 16,1 SOLUTIONS 1105...

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Unformatted text preview: 16,1 SOLUTIONS 1105 CHAPTER SIXTEEN lutians for Section 16.1 M k the values of the fuction on the plane, as shown in Figue 16.1, so that you can guess respectively at the smalest an~iarest values the fuction takes on each small rectangle. Lower sum = L f(Xi, Yi)D.XD.y = 4D.xD.y + 6D.xD.y + 3D.xD.y + 4D.xD.y = 17 D.xD.y = 17(0.1) (0.2) = 0.34. Upper sum = L f(Xi, Yi)D.XD.y = 7 D.xD.y + 10D.xD.y + 6D.xD.y + 8D.xD.y = 31D.xD.y = 31(0.1)(0.2) = 0.62. y l 3 2.4 5 4 f2.2 ,4 b.y = 0.2 12.0 l 1.0 6 8-- 1.1 10 1.2 ;. x !+b.x = 0.1-- Figure 16.1 :z In the subrectagle in the top left in Figure 16.4, it appears that f(x, y) has a maxum value of about 9. In the subrect- angle in the top middle, f(x, y) has a maxmum value of 10. Continuing in ths way, and multiplying by D.x and D.y, we have Overestiate = (9 + 10 + 12 + 7 + 8 + 10 + 5 + 7 + 8)(10)(5) = 3800. Simarly, we fid Underestiate = (7 + 7 + 8 + 4 + 5 + 7 + 1 + 3 + 6)(10)(5) = 2400. Thus, we expet that 2400:S i f(x,y)dA 'S 3800. 3. (a) If we tae the partion of R consisting of just R itself, we get Lower bound forintegral = mIRf . AR = 0 . (4 - 0)(4 - 0) = O. Simarly, we get Upper bound for integral = 'maxRf . AR = 4. (4 - 0)(4 - 0) = 64. 1106 Chapter Sixteen SOLUTIONS (b) Th """" "'ked to or ju" tho "PP"""'d low" "". Wo _"00 R i"oo '"b,,,,,,,, R,o,,) of widll 2 . hcght 2, whor ( ", b) ~ th low".to" 'om" of R, 0,')' Th '"""I" or thon R,o,o), 11',0), R,o,,), ,,d 11, (0,4) (0,0) (4,4) (4,0) Figure 16.2 Then we find the lower Sum Lower sum = AR(a,b) . Rmin 1= 4. (Min of Ion R(a,b)) (a,b) (a,b) (a,b) = 4 (Min of Ion R(a,b)) (a,b) = 4(1(0,0) + 1(2, 0) + 1(0, 2) + 1(2, 2)) =4(~+~+Vi+~) =8. Similarly, the upper sum is Upper sum = 4 (Max of Ion R(a,b)) (a,b) = 4(1(2,2) + 1(4, 2) + 1(2, 4) + 1(4, 4)) =4(~+v'+~+~) = 24 + 16V2 ~ 46.63. Th 0_ ""n i, "" "",- ""d th low" 'om ~ "" """",,"_, 00 wo ''' got . "'tt "li", by ,,,_ them to get 16 + 80 ~ 27.3. 4. (.) Wo fi, find"" 0"". "" nnde,,"_ of th in",gr, a'ing f.. ,nhre,,,,,. On tho fi, oobrelo (0 ~ x ~ 3, 0 ~ y S 4), th fu'"oo I( x, y) -. to have . m""inmm of 100 ""d . mininnnn of 79. Conlig in .. way, and using the fact that D.x = 3 and D.y = 4, we have Overestiate = (100 + 90 + 85 + 79)(3)(4) = 4248, and Underestimate = (79 + 68 + 61 + 55)(3)(4) = 3156. A better estiate of the integral is the average of the overestimate and the underestimate: . 4248 + 3156 Better estimate = 2 = 3702. (b) Th ""' ",no of I (x, y) on th, "'gioo ~ tho V"'ae of th ingr thvidcd by tho "" of th mgon. Sin ~, area of R is (6)(8) = 48, we approximate '. 1 l 1 Average value = - I(x, Y)dA ~ _ .3702 = 77.125. Area R 48 Wo "' in th tabto th tho "'"" of I (x, y) on th, "'gioo v"y "'!wren 55 ""d 100. '0 "" "=go ",no of77,t, i, ""nalo, , S. -"00 R inoo ,n""omg'" with th Ii",,, x ~ 0, x ~ 0.5, x ~ 1, x ~ 1.5. ""d x ~ 2"" th lin" Y ~ 0..," ~ 2, ~ 3. ""d ~ 4, Th wo have 16 ,nbretal", _ of whioh wo doo", R,o,.), whor ( a, b) i, th ~ the lower-left comer of the subrectagle. . 16.1 SOLUTIONS16....
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ECOPY2_LDAPMAIL_12092008-095656 - 16,1 SOLUTIONS 1105...

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