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15.3withScannedExamples

# 15.3withScannedExamples - 15.3 DOUBLE INTEGRALS OVER...

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Unformatted text preview: 15.3: DOUBLE INTEGRALS OVER GENERAL REGIONS KIAM HEONG KWA 1. DOUBLE INTEGRALS OVER GENERAL REGIONS Let f(:c,y) be a function deﬁned on a bounded region D in the sense that D can be enclosed in a rectangular region R. We deﬁne the double integral of f over D by setting (1.1) //;)f(m,y)dA=//12F(x,y)dA provided the integral on the right—hand side exists, where F is the function (1.2) may) : {1605,13 if (azy) E D, 0 if (33,31) 6 R\D. Here R\D means the set of all points that lie in R but not in D. The integral given in (1.1) does not exist without further conditions on the function f and its domain D. We shall give a very restrictive set of conditions. More general conditions are beyond the scope of this course. A plane region D1 is said to be of type I if it lies between the graphs of two continuous functions gl and 92(x) of a). That is, D: = {(96,211) E R2la S x S b,gi(m) S y S 92%}, where [a, b] denotes a closed and bounded interval. In this case, the double integral of f over D1 exists and (1.3) //DI f(rv,y)dA= b f(\$,y)dyd\$ Date: October 6, 2010. 2 KIAM HEONG KWA provided f is continuous on D1. Similarly, a plane region DH is said to be of type II if it lies between the graphs of two continuous functions h1(y) and h; (y) of y. That is, DH = {(m,y) e Rzlc s y s d,h1(y) s so 3 h2(y)}, Where [c,d] denotes a closed and bounded interval. In this case, the double integral of f over DH exists and (1.4) f/D“ fem/MA = d /h f<x,y)dxdy provided f is continuous on D”. Some Properties of Double Integrals. We assume that all the following integrals exist and D is a bounded plane region. 0 Linearity: For any real constant c, fi/le(x,y)+g(a:,y)]dA=//D f(x,y)dA+//D g(x,y)dA, //DC-f(\$,y)dA=C//Df(a:,y)dA. o Additivity: If D is a disjoint union of two plane regions D1 and D2, then M ﬂat y) M = / DI mad/1 + / D2 rm) am. 0 If A(D) is the area of D, then A(D) = 1dA. o Monotonicity: If g(x,y) S f (cc,y) for all (3;,y) E D, then //D 9(m,y)dAS//D f(:v,y)dA- In particular, if m S f (33,?!) S M for all (x, y) E D, then m.A<D> s mad/is M-A<D). D Example 1. Use the monotonicity of integrals to estimate ffD #323 + y3 dA, where D = [0,1] X [0,1]. Solution. Obviously, the absolute maximum and the absolute minimum of «:33 + if over D are 0 and V? respectively. Hence 03 x/ac3+y3_<_\/§ 15.3: DOUBLE INTEGRALS OVER GENERAL REGIONS on D. Therefore 0=o-A(D)g// x/x3+y3dAgx/§-A(D)=\/§. D 4" x4 xco DJXT—H) “MESH—ENE} ’ 3517f lo ,, , “299) mm 3605133me M‘hm =90 En) vokume 0'? ‘Me Mid Mckr {14f B—raxﬁé,‘ QM abode 7““ ream Banal )C:33 . X22 2 2:3 «A 2 O. Y4- + ” we» gm me ww‘b {f C¥’*°b‘>4\/_. w m”: ﬁne pkmas 2=><j and 2': O‘ W #_ ‘VH—ﬁqce Me Vo\u%€ 0"? “We {\S {x 1 __ i _E,_fﬂ,:g..£l2><_+\"ﬂ _ _ o ___ #__ # -_.7,.,.7 _. _,.--‘!I7h.eg%mins&goh4 j;‘ ma - — w~ _w_,_ -— BL? um - _ A , ,, .emdeéfg.’ Pauab bio Ld‘hm“ _. . ,..;_.7:e;;§ft2>j-3n_d ﬁt 913w, X59, ; maai‘zui’fﬁéqm H _ pm wﬁyzsfm‘ V 751% wékecoﬁaw " n 1/ roof? T: _ \2 c , y*2*1)% " x" ‘ ” " " ’U ‘ bUS—e ‘; >90 — / Nd‘ii;§,5“3f’§‘fwiz‘?fua {HS _ ‘ f 7 v M @454. §’f‘f5’,‘l~>?=,°<”8‘f‘7‘”ej ‘ E mg H 22: Rwy”- ZD 6m; 4 H‘ ’ ‘ L5 W 626 Al :91}: VERA/he 8&8 baundeol b5 . War yak—A MAW Planes x ffb Kw gum—E r Tm: minim“: “ * "* ...
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