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# 15.7withScannedExamples - 15.7 15.8 TRIPLE INTEGRALS IN...

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Unformatted text preview: 15.7 / 15.8: TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICIAL COORDINATES KIAM HEONG KWA 1. TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES For a continuous function f on a bounded solid region E = {QM/,2) E R3l(x,y) 6 D,U1(rv,y) S 2 S u2(x,y)} of type 1 in the three-dimensional Euclidean space that projects onto a plane region D given in polar coordinates by D = {(739) 6 WM 3 6 s ﬁ,h1(0)s r 3 152(6)}, where 0 S ,8 — oz < 27r and h1(6) and h2(0) are continuous functions, it may be easier to evaluate the integral // E f(a:,y,z>dv = //D [ f(m,y,z)dz in terms of cylindrical coordinates. dA Recall that the cylindrical coordinates (r, 0, z) of a point are related to its rectangular coordinates (:r, y, 2) by the relations at = r0036, y: rsinﬁ, z = 2. In these coordinates, (1.1) ﬂ hz (0) uz (1* cos 0,1" sin 0) f(m,y,z)dV=/ / / f(rcos0,rsin6)rdzdrd0. E a h1(0) u1(r cos 0,1“ sin 0) 2. TRIPLE INTEGRALS IN SPHERICAL COORDINATES For a continuous function f on a spherical wedge, i.e., a solid region of the form E={(p,0,¢) 6R3Ia3p3b,6l 36361451 @952}, Date: October 16, 2010. 2 KIAM HEONG KWA whereOS as b, 03 02—01 < 27r, 0 S ¢2—¢1 <7r, and (p,¢9,¢) are the spherical coordinates related to the rectangular coordinates by the relations a: = psin¢cos9, y = psingbsinﬁ, z = pcosqb,‘ it can be shown that (2.1) f(:c,y,z) dV ¢2 92 b =/ / / f(psin¢>cos0,psinqt>sint9,pcosqﬁ)p2 sinqbddedqb. 451 91 a It should be noted that though we have deﬁned triple integrals by dividing solids into small rectangular boxes, the same result can be achieved by dividing a solid into small spherical wedges. This follows from a more general theory that we shall discuss in the next section. To see how triple integrals can be deﬁned in spherical coordinates directly, interested readers are referred to p. 1007 of the text. 89: \5. 3}: Triple InﬁmQ-P fin Qﬁh‘ad‘d CKQ CcmdiMd-es Rye-52‘ 13.: W “ " S’PR‘Ww' QWTMAU _______ﬂ-,4:’>‘r — . (51;. )9 GW E L: mam m Me‘Rfﬁ’ 90 Gwen EismkcseA — We Wm {40 owl 21-. x+oﬂ+a-__ '(W mghﬂSWs XZHO 2:24, DNA, : —er~’—°L¢—,——g_——~-~—~— - . 19. 7(JQ5C‘ AIDA I'A- / . ,w e fleﬁﬁ gm! A I-ﬂlh“? —— Cr C‘59)?4' CrmwiCrghg’S’ y ll I m w W * II x - E D~ F: i=0 _2__ 3:" g r((§bB+5(“e +5 ‘ U I! ! ﬁ4< 'I n "I u I - I ‘ \ /6'?~ 7 m“ _ V ‘ 7 901 PM {Me Vasle 3’?“ ﬁg 32.2215! E 7 f“ = «gorse ' 4M9!“ ‘(fs er’V/f. 96W ( \mém/ 7 Xl—lcjafi 01M, Jib? 509394 = 7:0?“ ?> 9) b :4 4c: —— 1 €1.55 ' I d“ l . 24 C00 PM! ‘A/K vbtm a? MRS‘MA E 5 ‘ rater ; ' /" /5Fr" MOW {\= Cuts an. #3 :__I (=0 we 9mm a 0+ < me = _1-| a 1 Ill “X 4 A w = (F 03’?” PM?" A I I v . “1-:O Xc~ : F A —- 5 ° W wmdwawwm A- E III NM acm— .. ' ‘ :3 .g :0 .l f ":0 15.2.1 Echbe‘mw weaves #2 VCS)‘: d m w =4 m mm me =. l ' __ J. Skg 5° 2 V l as w. , m 9e OA~UAQ g-gian«g I! ME , . 27F + v 8‘ = .— rsmms e m ' rammed ~— ‘— a— 0 [9(8'0 ﬁnal ‘Um? d skiﬂi 8+ [tea W's Mspkam K14? L-E Elt“: AW": Wu: x. 49km?) _ :2 m5 Edd -éh42 Cam i—tJ x24 'L . '. ‘ﬂkﬁw-l I‘m—I .— ’1 "' 751': q' I ...
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15.7withScannedExamples - 15.7 15.8 TRIPLE INTEGRALS IN...

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