ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 73 CA tive -is psses throgh the point where the prime meridin (the meridin throgh Greenwich, Englnd) intersects the eqtor. hen the ltitde of P is nd the longitde is. Find the gret-circle distnce from Los Angeles (lt. 34.6 N, long. 8.5 W) to Montrél (lt. 45.5 N, long. 73.6 W). ke the rdis of the Erth to be 396 mi. (A gret circle is the circle of intersection of sphere nd plne throgh the center of the sphere.) 36 5 sin m sin n 9 39. he srfces hve been sed s models for tmors. he bmp sphere with m 6 nd n 5 is shown. Use compter lgebr sstem to find the volme it encloses. 4. how tht e z d d dz s z (he improper triple integrl is defined s the limit of triple integrl over solid sphere s the rdis of the sphere increses indefinitel.) 4. () Use clindricl coordintes to show tht the volme of the solid bonded bove b the sphere r z nd below b the cone z r cot (or ), where, is (b) Dedce tht the volme of the sphericl wedge given b,, is (c) Use the Men Vle heorem to show tht the volme in prt (b) cn be written s where lies between nd, lies between nd,,, nd. V 3 V 3 cos 3 3 V 3 cos cos sin.8 CHANGE OF VAIABLE IN MULIPLE INEGAL In one-dimensionl clcls we often se chnge of vrible ( sbstittion) to simplif n integrl. B reversing the roles of nd, we cn write the bstittion le (5.5.6) s b f d d f tt d c where t nd tc, b td. Another w of writing Forml is s follows: A chnge of vribles cn lso be sefl in doble integrls. We hve lred seen one emple of this: conversion to polr coordintes. he new vribles r nd re relted to the old vribles nd b the eqtions nd the chnge of vribles forml (.3.) cn be written s b f, da r f d d r cos c f d d d r sin f r cos, r sin rdrd where is the region in the -plne tht corresponds to the region in the -plne.
74 CHAPE MULIPLE INEGAL More generll, we consider chnge of vribles tht is given b trnsformtion from the v-plne to the -plne:, v, where nd re relted to nd v b the eqtions 3 t, v h, v or, s we sometimes write,, v, v We sll ssme tht is C trnsformtion, which mens tht t nd h hve continos first-order prtil derivtives. A trnsformtion is rell jst fnction whose domin nd rnge re both sbsets of. If, v,, then the point, is clled the imge of the point, v. If no two points hve the sme imge, is clled one-to-one. Figre shows the effect of trnsformtion on region in the v-plne. trnsforms into region in the -plne clled the imge of, consisting of the imges of ll points in. (, )! (, ) FIGUE FIGUE (, ) (, ) = - 4 (_, ) (, ) (, ) =- 4 (, ) If is one-to-one trnsformtion, then it hs n inverse trnsformtion from the -plne to the v-plne nd it m be possible to solve Eqtions 3 for nd v in terms of nd : G, v H, V EXAMPLE A trnsformtion is defined b the eqtions v v Find the imge of the sqre, v, v. OLUION he trnsformtion mps the bondr of into the bondr of the imge. o we begin b finding the imges of the sides of. he first side,, is given b v. (ee Figre.) From the given eqtions we hve,, nd so. hs is mpped into the line segment from, to, in the -plne. he second side,, is v nd, ptting in the given eqtions, we get v v Eliminting v, we obtin 4 4
ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 75 which is prt of prbol. imilrl, 3 is given b v, whose imge is the prbolic rc 5 4 Finll, is given b whose imge is v 4 v,, tht is,. (Notice tht s we move rond the sqre in the conterclockwise direction, we lso move rond the prbolic region in the conterclockwise direction.) he imge of is the region (shown in Figre ) bonded b the -is nd the prbols given b Eqtions 4 nd 5. Now let s see how chnge of vribles ffects doble integrl. We strt with smll rectngle in the v-plne whose lower left corner is the point, v nd whose dimensions re nd v. (ee Figre 3.) = r (, ) Î (, ) Î = (, ) r (, ) FIGUE 3 he imge of is region in the -plne, one of whose bondr points is,, v. he vector r, v t, v i h, v j is the position vector of the imge of the point, v. he eqtion of the lower side of is v v, whose imge crve is given b the vector fnction r, v. he tngent vector t, to this imge crve is r t, v i h, v j i j r (, +Î) r (, ) r ( +Î, ) FIGUE 4 imilrl, the tngent vector t, to the imge crve of the left side of (nmel, ) is r v t v, v i h v, v j We cn pproimte the imge region b prllelogrm determined b the secnt vectors r, v r, v b r, v v r, v shown in Figre 4. Bt r, v r, v r lim l v i v j
76 CHAPE MULIPLE INEGAL nd so r, v r, v r Îr r (, ) FIGUE 5 imilrl r, v v r, v vr v his mens tht we cn pproimte b prllelogrm determined b the vectors r nd v r v. (ee Figre 5.) herefore, we cn pproimte the re of b the re of this prllelogrm, which, from ection.4, is 6 Compting the cross prodct, we obtin r v r v r r v v r r v i j k v v v v k v v k he determinnt tht rises in this clcltion is clled the Jcobin of the trnsformtion nd is given specil nottion. he Jcobin is nmed fter the Germn mthemticin Crl Gstv Jcob Jcobi (84 85). Althogh the French mthemticin Cch first sed these specil determinnts involving prtil derivtives, Jcobi developed them into method for evlting mltiple integrls. 7 DEFINIION he Jcobin of the trnsformtion given b t, v nd h, v is,, v v v v v With this nottion we cn se Eqtion 6 to give n pproimtion to the re A of : 8 where the Jcobin is evlted t, v. Net we divide region in the v-plne into rectngles ij nd cll their imges in the -plne. (ee Figre 6.) ij, A, v v ij Î Î ij ( i, j ) ( i, j) FIGUE 6
ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 77 f Appling the pproimtion (8) to ech ij, we pproimte the doble integrl of over s follows: f, da m m n i j n i j f i, j A, f t i, v j, h i, v j, v v where the Jcobin is evlted t i, v j. Notice tht this doble sm is iemnn sm for the integrl, f t, v, h, v, v d dv he foregoing rgment sggests tht the following theorem is tre. (A fll proof is given in books on dvnced clcls.) 9 CHANGE OF VAIABLE IN A DOUBLE INEGAL ppose tht is C trnsformtion whose Jcobin is nonzero nd tht mps region in the v- plne onto region in the -plne. ppose tht f is continos on nd tht nd re tpe I or tpe II plne regions. ppose lso tht is one-toone, ecept perhps on the bondr of. hen f, da, f, v,, v, v d dv å r= = =å r=b heorem 9 ss tht we chnge from n integrl in nd to n integrl in nd v b epressing nd in terms of nd v nd writing, da, v d dv = r= å FIGUE 7 he polr coordinte trnsformtion =å b r=b r Notice the similrit between heorem 9 nd the one-dimensionl forml in Eqtion. Insted of the derivtive dd, we hve the bsolte vle of the Jcobin, tht is,,, v. As first illstrtion of heorem 9, we show tht the forml for integrtion in polr coordintes is jst specil cse. Here the trnsformtion from the -plne to the -plne is given b nd the geometr of the trnsformtion is shown in Figre 7. mps n ordinr rectngle in the -plne to polr rectngle in the -plne. he Jcobin of is r, r, r r tr, r cos cos sin hr, r sin r r sin r cos r cos r sin r
78 CHAPE MULIPLE INEGAL hs heorem 9 gives f, d d which is the sme s Forml.3.., f r cos, r sin r, dr d b f r cos, r sin rdrd V EXAMPLE Use the chnge of vribles v, v to evlte the integrl da, where is the region bonded b the -is nd the prbols 4 4 nd 4 4,. OLUION he region is pictred in Figre (on pge 74). In Emple we discovered tht, where is the sqre,,. Indeed, the reson for mking the chnge of vribles to evlte the integrl is tht is mch simpler region thn. First we need to compte the Jcobin:,, v v v v v 4 4v herefore, b heorem 9, da 8, v, v da 3 v v 3 d dv 8 v4 v d dv [ 4 4 v v 3 ] dv v 4v 3 dv [v v 4 ] NOE Emple ws not ver difficlt problem to solve becse we were given sitble chnge of vribles. If we re not spplied with trnsformtion, then the first step is to think of n pproprite chnge of vribles. If f, is difficlt to integrte, then the form of f, m sggest trnsformtion. If the region of integrtion is wkwrd, then the trnsformtion shold be chosen so tht the corresponding region in the v-plne hs convenient description. EXAMPLE 3 Evlte the integrl e da, where is the trpezoidl region with vertices,,,,,, nd,. OLUION ince it isn t es to integrte sggested b the form of this fnction: e, we mke chnge of vribles v hese eqtions define trnsformtion from the -plne to the v-plne. heorem 9 tlks bot trnsformtion from the v-plne to the -plne. It is
ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 79 obtined b solving Eqtions for nd : he Jcobin of is v v,, v v v (_, ) = (, ) =_ = (_, ) (, ) =! -= _ -= _ o find the region in the v-plne corresponding to, we note tht the sides of lie on the lines nd, from either Eqtions or Eqtions, the imge lines in the v-plne re hs the region is the trpezoidl region with vertices,,,,,, nd, shown in Figre 8. ince heorem 9 gives v, v e da v v v v v, v v, e v, v d dv e v ( ) d dv v [ve v v ] v dv FIGUE 8 e e v dv 3 4e e IPLE INEGAL here is similr chnge of vribles forml for triple integrls. Let be trnsformtion tht mps region in vw-spce onto region in z-spce b mens of the eqtions t, v, w h, v, w z k, v, w he Jcobin of is the following 3 3 determinnt:,, z, v, w z v v z v w w z w
7 CHAPE MULIPLE INEGAL Under hpotheses similr to those in heorem 9, we hve the following forml for triple integrls: 3 f,, z dv f, v, w,, v, w, z, v, w,, z, v, w d dv dw V EXAMPLE 4 Use Forml 3 to derive the forml for triple integrtion in sphericl coordintes. OLUION Here the chnge of vribles is given b sin cos sin sin z cos We compte the Jcobin s follows: sin sin sin cos,, z,, sin sin sin cos cos cos sin sin sin cos cos cos cos sin cos cos cos sin sin sin sin cos sin sin sin sin sin cos cos sin cos sin sin cos cos sin cos sin sin sin ince, we hve sin. herefore nd Forml 3 gives sin sin cos f,, z dv,, z,, sin sin f sin cos, sin sin sin sin, cos sin d d d which is eqivlent to Forml.7.3.
ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 7.8 EXECIE 6 Find the Jcobin of the trnsformtion... 4v, v, 3 v v 3., v v v 4. sin, cos 5. v, vw, z w 6. e v, e v, z e vw 7 Find the imge of the set nder the given trnsformtion. 7., v 3, v ; 3v, v 8. is the sqre bonded b the lines,, v, v ; v, v 9. is the tringlr region with vertices,,,,, ;, v. is the disk given b v ;, bv 6 Use the given trnsformtion to evlte the integrl.. 3 da, where is the tringlr region with vertices,,,, nd, ; v, v. 4 8 da, where is the prllelogrm with vertices, 3,, 3, 3,, nd, 5; 4 v, 4v 3 3. da, where is the region bonded b the ellipse 9 4 36;, 3v 4. da, where is the region bonded b the ellipse ; s s3 v, s s3 v 5. da, where is the region in the first qdrnt bonded b the lines nd 3 nd the hperbols, 3; v, v ; 6. da, where is the region bonded b the crves,,, ;, v. Illstrte b sing grphing clcltor or compter to drw. 7. () Evlte, where E is the solid enclosed b the ellipsoid b z c. Use the trnsformtion, bv, z cw. (b) he Erth is not perfect sphere; rottion hs reslted in flttening t the poles. o the shpe cn be pproimted b n ellipsoid with b 6378 km nd c 6356 km. Use prt () to estimte the volme of the Erth. 8. Evlte E dv, where E is the solid of Eercise 7(). 9 3 Evlte the integrl b mking n pproprite chnge of vribles. 9., where is the prllelogrm enclosed b 3 da the lines, 4, 3, nd 3 8. e da, where is the rectngle enclosed b the lines,,, nd 3. cos da, where is the trpezoidl region with vertices,,,,,, nd,. sin9 4 da, where is the region in the first qdrnt bonded b the ellipse 9 4 3. e da, where is given b the ineqlit E dv 4. Let f be continos on, nd let be the tringlr region with vertices,,,, nd,. how tht f da f d