The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or more vribles. Crtesin produts We will find the following nottion useful. Given two sets of rel numbers A nd B, we define the Crtesin produt of A nd B to be the set For exmple, if A {,, } nd B {5, 6}, then A B {x, y) : x A, y B}..6.) A B {, 5),, 6),, 5),, 6),, 5),, 6)}. In prtiulr, if < b, < d, A [, b], nd B [, d], then A B [, b] [, d] is the losed retngle {x, y) : x b, y d}, s shown in Figure.6.. d b Figure.6. The losed retngle [, b] [, d] More generlly, given rel numbers i < b i, i,,,..., n, we my write [, b ] [, b ] [ n, b n ] Copyright by n Sloughter
efinite Integrls Setion.6 for the losed retngle nd for the open retngle efinite integrls on retngles Given < b nd < d, let {x, x,..., x n ) : i x i b i, i,,..., n}, b ), b ) n, b n ) {x, x,..., x n ) : i < x i < b i, i,,..., n}. [, b] [, d] nd suppose f : R R is defined on ll of. Moreover, we suppose f is bounded on, tht is, there exist onstnts m nd M suh tht m fx, y) M for ll x, y) in. In prtiulr, the Extreme Vlue Theorem implies tht f is bounded on if f is ontinuous on. Our definition of the definite integrl of f over the retngle will follow the definition from one-vrible lulus. Given positive integers m nd n, we let P be prtition of [, b] into m intervls, tht is, set P {x, x,..., x m } where x < x < < x m b,.6.) nd we let Q be prtition of [, d] into n intervls, tht is, set Q {y, y,..., y n } where y < y < < y n d..6.) We will let P Q denote the prtition of into mn retngles ij [x i, x i ] [y j, y j ],.6.4) where i,,..., m nd j,,..., n. Note tht ij hs re x i y j, where nd An exmple is shown in Figure.6.. x i x i x i.6.5) y j y j y j..6.6) d 4 b Figure.6. A prtition of retngle [, b] [, d]
Setion.6 efinite Integrls Now let m ij be the lrgest rel number with the property tht m ij fx, y) for ll x, y) in ij nd M ij be the smllest rel number with the property tht fx, y) M ij for ll x, y) in ij. Note tht if f is ontinuous on, then m ij is simply the minimum vlue of f on ij nd M ij is the mximum vlue of f on ij, both of whih re gurnteed to exist by the Extreme Vlue Theorem. If f is not ontinuous, our ssumption tht f is bounded nevertheless gurntees the existene of the m ij nd M ij, lthough the justifition for this sttement lies beyond the sope of this book. We my now define the lower sum, Lf, P Q), for f with respet to the prtition P Q by m n Lf, P Q) m ij x i y j.6.7) i j nd the upper sum, Uf, P Q), for f with respet to the prtition P Q by Uf, P Q) m n M ij x i y j..6.8) i j Geometrilly, if fx, y) for ll x, y) in nd V is the volume of the region whih lies beneth the grph of f nd bove the retngle, then Lf, P Q) nd Uf, P Q) represent lower nd upper bounds, respetively, for V. See Figure.6. for n exmple of one term of lower sum). Moreover, we should expet tht these bounds n be mde rbitrrily lose to V using suffiiently fine prtitions P nd Q. In prt this implies tht we my hrterize V s the only rel number whih lies between Lf, P Q) nd Uf, P Q) for ll hoies of prtitions P nd Q. This is the bsis for the following definition. efinition Suppose f : R R is bounded on the retngle [, b] [, d]. With the nottion s bove, we sy f is integrble on if there exists unique rel number I suh tht Lf, P Q) I Uf, P Q).6.9) for ll prtitions P of [, b] nd Q of [, d]. If f is integrble on, we ll I the definite integrl of f on, whih we denote I fx, y)dxdy..6.) Geometrilly, if fx, y) for ll x, y) in, we my think of the definite integrl of f on s the volume of the region in R whih lies beneth the grph of f nd bove the retngle. Other interprettions inlude totl mss of the retngle if fx, y) represents the density of mss t the point x, y)) nd totl eletri hrge of the retngle if fx, y) represents the hrge density t the point x, y)). Exmple Suppose fx, y) x + y nd [, ] [, ]. If we let P {,, }
4 efinite Integrls Setion.6 y x 7.5 z 5.5 Figure.6. Grph of fx, y) x + y showing one term of lower sum nd Q {,,, }, then the minimum vlue of f on eh retngle of the prtition P Q ours t the lower left-hnd orner of the retngle nd the mximum vlue of f ours t the upper righthnd orner of the retngle. See Figure.6. for piture of one term of the lower sum. Hene Lf, P Q) f, ) ) + f, + f, ) ) + f, + f, ) ) + f, + 8 + + 5 8 + + 7 8 4 8 5.75 nd Uf, P Q) f ), + f, ) ) + f, + f, ) + f, ) + f, )
Setion.6 efinite Integrls 5 5 8 + + 7 8 + 5 + 7 8 + 5 7 8 5.875. We will see below tht the ontinuity of f implies tht f is integrble on, so we my onlude tht 5.75 x + y )dxdy 5.875. Exmple Suppose k is onstnt nd fx, y) k for ll x, y) in the retngle [, b] [, d]. The for ny prtitions P {x, x,..., x m } of [, b] nd Q {y, y,..., y n } of [, d], m ij k M ij for i,,..., m nd j,,..., n. Hene Lf, P Q) Uf, P Q) m n k x i y j k Hene f is integrble nd fx, y)dxdy i j m i j n x i y j k re of ) kb )d ). kdxdy kb )d ). Of ourse, geometrilly this result is sying tht the volume of box with height k nd bse is k times the re of. In prtiulr, if k we see tht dxdy re of. Exmple If [, ] [, ], then 5dxdy 5 ) + ). The properties of the definite integrl stted in the following proposition follow esily from the definition, lthough we will omit the somewht tehnil detils. Proposition Suppose f : R R nd g : R R re both integrble on the retngle [, b] [, d] nd k is slr onstnt. Then fx, y) + gx, y))dxdy fx, y)dxdy + gx, y)dxdy,.6.)
6 efinite Integrls Setion.6 kfx, y)dxdy k fx, y)dxdy,.6.) nd, if fx, y) gx, y) for ll x, y) in, fx, y)dxdy gx, y)dxdy..6.) Our definition does not provide prtil method for determining whether given funtion is integrble or not. A omplete hrteriztion of integrbility is beyond the sope of this text, but we shll find one simple ondition very useful: if f is ontinuous on n open set ontining the retngle, then f is integrble on. Although we will not ttempt full proof of this result, the outline is s follows. If f is ontinuous on [, b] [, d] nd we re given ny ɛ >, then it is possible to find prtitions P of [, b] nd Q of [, d] suffiiently fine to gurntee tht if x, y) nd u, v) re points in the sme retngle ij of the prtition P Q of, then fx, y) fu, v) < ɛ b )d )..6.4) Note tht this is not diret onsequene of the ontinuity of f, but follows from slightly deeper property of ontinuous funtions on losed bounded sets known s uniform ontinuity.) It follows tht if m ij is the minimum vlue nd M ij is the mximum vlue of f on ij, then Uf, P Q) Lf, P Q) < ɛ. m i j m i j m i j n M ij x i y j m i j n M ij m ij ) x i y j n ɛ b )d ) ɛ b )d ) x i y j m i j n x i y j ɛ b )d ) b )d ) n m ij x i y j.6.5) It now follows tht we my find upper nd lower sums whih re rbitrrily lose, from whih follows the integrbility of f. Theorem If f is ontinuous on n open set ontining the retngle, then f is integrble on.
Setion.6 efinite Integrls 7 y x 7.5 z 5.5 Figure.6.4 A slie of the region beneth fx, y) x + y with re α) Exmple If fx, y) x + y, then f is ontinuous on ll of R. Hene f is integrble on [, ] [, ]. Iterted integrls Now suppose we hve retngle [, b] [, d] nd ontinuous funtion f : R R suh tht fx, y) for ll x, y) in. Let B {x, y, z) : x, y), z fx, y)}..6.6) Then B is the region in R bounded below by nd bove by the grph of f. If we let V be the volume of B, then V fx, y)dxdy..6.7) However, there is nother pproh to finding V. If, for every y d, we let αy) b fx, y)dx,.6.8) then αy) is the re of slie of B ut by plne orthogonl to both the xy-plne nd the yz-plne nd pssing through the point, y, ) on the y-xis see Figure.6.4 for n exmple). If we let the prtition Q {y, y,..., y n } divide [, d] into n intervls of equl length y, then we my pproximte V by n αy i ) y..6.9) j
8 efinite Integrls Setion.6 Tht is, we my pproximte V by sliing B into slbs of thikness y perpendiulr to the yz-plne, nd then summing pproximtions to the volume of eh slb. As n inreses, this pproximtion should onverge to V ; t the sme time, sine.6.9) is right-hnd rule pproximtion to the definite integrl of α over [, d], the sum should onverge to d αy)dy s n inreses. Tht is, we should hve V lim n j n αy i ) y d αy)dy d ) b fx, y)dx dy..6.) Note tht the expression on the right-hnd side of.6.) is not the definite integrl of f over, but rther two suessive integrls of one vrible. Also, we ould hve reversed our order nd first integrted with respet to y nd then integrted the result with respet to x. efinition Suppose f : R R is defined on retngle [, b] [, d]. The iterted integrls of f over re d b fx, y)dxdy d ) b fx, y)dx dy.6.) nd b d fx, y)dydx b ) d fx, y)dy dx..6.) In the sitution of the preeding prgrph, we should expet the iterted integrls in.6.) nd.6.) to be equl sine they should both equl V, the volume of the region B. Moreover, sine we lso know tht V fx, y)dxdy, the iterted integrls should both be equl to the definite integrl of f over. These sttements my in ft be verified s long s f is integrble on nd the iterted integrls exist. In this se, iterted integrls provide method of evluting double integrls in terms of integrls of single vrible for whih we my use the Fundmentl Theorem of Clulus).
Setion.6 efinite Integrls 9 y x 7.5 z 5.5 Figure.6.5 Region beneth fx, y) x + y over the retngle [, ] [, ] Fubini s Theorem for retngles) Suppose f is integrble over the retngle [, b] [, d]. If exists, then If exists, then d b fx, y)dxdy b d fx, y)dxdy fx, y)dxdy d b fx, y)dydx b d fx, y)dxdy..6.) fx, y)dxdy..6.4) Exmple To find the volume V of the region beneth the grph of fx, y) x + y nd over the retngle [, ] [, ] s shown in Figure.6.5), we ompute V x + y )dxdy x + y )dxdy
efinite Integrls Setion.6 ) x + xy dy ) + y dy ) y + y + 9. We ould lso ompute the iterted integrl in the other order: V x + y )dxdy x + y )dydx x y + y x + 9)dx x + 9y) + 9. ) dx Exmple If [, ] [, ], then x ydxdy x ydydx x y dx x dx x 6 8 6 6 7 6. efinite integrls on other regions Integrls over intervls suffie for most pplitions of funtions of single vrible. However, for funtions of two vribles it is importnt to onsider integrls on regions other thn retngles. To extend our definition, onsider funtion f : R R defined on bounded region. Let be retngle ontining nd, for ny x, y) in, define f x, y) { fx, y), if x, y),, if x, y) /..6.5) In other words, f is identil to f on nd t ll points of outside of. Now if f is integrble on, nd sine the the region where f is should ontribute nothing
Setion.6 efinite Integrls y β x) d x γ y) x δ y) y α x) b Figure.6.6 Regions of Type I nd Type II to the vlue of the integrl, it is resonble to define the integrl of f over to be equl to the integrl of f over. efinition Suppose f is defined on bounded region of R nd let be ny retngle ontining. efine f s in.6.5). We sy f is integrble on if f is integrble on, in whih se we define fx, y)dxdy f x, y)dxdy..6.6) Note tht the integrbility of f on region depends not only on the nture of f, but on the region s well. In prtiulr, even if f is ontinuous on n open set ontining, it my still turn out tht f is not integrble on beuse of the omplited nture of the boundry of. Fortuntely, there re two bsi types of regions whih our frequently nd to whih our previous theorems generlize. efinition We sy region in R is of Type I if there exist rel numbers < b nd ontinuous funtions α : R R nd β : R R suh tht αx) βx) for ll x in [, b] nd {x, y) : x b, αx) y βx)}..6.7) We sy region in R is of Type II if there exist rel numbers < d nd ontinuous funtions γ : R R nd δ : R R suh tht γy) δy) for ll y in [, d] nd {x, y) : y d, γy) x δy)}..6.8) Figure.6.6 shows typil exmples of regions of Type I nd Type II. Exmple If is the tringle with verties t, ),, ), nd, ), then {x, y) : x, y x}. Hene is Type I region with αx) nd βx) x. Note tht we lso hve {x, y) : y, y x }, so is lso Type II region with γy) y nd δy). See Figure.6.7.
efinite Integrls Setion.6.8.5.6.4. - -.5.5 -.5..4.6.8 - Figure.6.7 Two regions whih re of both Type I nd Type II Exmple The losed disk {x, y) : x + y } is both region of Type I, with {x, y) : x, x y x }, nd region of Type II, with {x, y) : y, y x x }. See Figure.6.7. Exmple Let be the region whih lies beneth the grph of y x nd bove the intervl [, ] on the x-xis. Then {x, y) : x, y x }, so is region of Type I. However, is not region of Type II. See Figure.6.8. Theorem If is region of Type I or region of Type II nd f : R R is ontinuous on n open set ontining, then f is integrble on. Fubini s Theorem for regions of Type I nd Type II) Suppose f : R R is integrble on the region. If is region of Type I with {x, y) : x b, αx) y βx)} nd the iterted integrl b βx) αx) fx, y)dydx
Setion.6 efinite Integrls.5.5 -.5 - -.5.5.5 Figure.6.8 A region whih is of Type I but not of Type II exists, then If is region of Type II with nd the iterted integrl exists, then fx, y)dxdy b βx) αx) {x, y) : y d, γy) x δy)} d δy) γy) fx, y)dxdy fx, y)dxdy d δy) γy) fx, y)dydx..6.9) fx, y)dydx..6.) Exmple Let be the tringle with verties t, ),, ), nd, ), s in the exmple bove. Expressing s region of Type I, we hve xydxdy x xydydx xy x dx x dx x4 8 8. Sine is lso region of Type II, we my evlute the integrl in the other order s well, obtining xydxdy y xydxdy x y dy y ) ) y y y dy 4 y4 8 8.
4 efinite Integrls Setion.6.8.6.4...4.6.8 Figure.6.9 The region {x, y) : x, x y } In the lst exmple the hoie of integrtion ws not too importnt, with the first order being perhps slightly esier thn the seond. However, there re times when the hoie of the order of integrtion hs signifint effet on the ese of integrtion. Exmple Let {x, y) : x, x y } see Figure.6.9). Sine is both of Type I nd of Type II, we my evlute e y dxdy either s or s y x e y dydx e y dxdy. The first of these two iterted integrls requires integrting gy) e y ; however, we my evlute the seond esily: y e y dxdy e y dxdy xe y y dy y e y dy e y e ).
Setion.6 efinite Integrls 5-4 - y z - - x Figure.6. Region bounded by z 4 x y nd the xy-plne Exmple Let V be the volume of the region lying below the prboloid P with eqution z 4 x y nd bove the xy-plne see Figure.6.). Sine the surfe P intersets the xy-plne when 4 x y, tht is, when x + y 4, V is the volume of the region bounded bove by the grph of fx, y) 4 x y nd below by the region {x, y) : x + y 4}. If we desribe s Type I region, nmely, then we my ompute V {x, y) : x, 4 x y 4 x }, 4 x y )dxdy 4 x 4 x y )dydx 4 x ) 4y x y y 4 x 4 x dx
6 efinite Integrls Setion.6 4 8 4 x x 4 x ) 4 x ) dx 4 x ) 4 x ) 4 x ) dx 4 x ) dx. Using the substitution x sinθ), we hve dx osθ)dθ, nd so V 4 4 64 64 6 6 6 6 8π. π π π π 4 x ) dx 4 4 sin θ)) osθ)dθ θ π π os 4 θ)dθ ) + osθ) dθ + osθ) + os θ))dθ π + θ π + π ) + sinθ) π π π π + + 8 sin4θ) π π π + os4θ) ) dθ ) Integrls of funtions of three or more vribles We will now sketh how to extend the definition of the definite integrl to higher dimensions. Suppose f : R n R is bounded on n n-dimensionl losed retngle [, b ] [, b ] [ n, b n ]. Let P, P,..., P n prtition the intervls [, b ], [, b ],..., [ n, b n ] into m, m,..., m n, respetively, intervls, nd let P P P n represent the orresponding prtition of into m m m n n-dimensionl losed retngles i i i n. If m i i i n is the lrgest rel number suh tht m i i i n fx) for ll x in i i i n nd M i i i n is the smllest
Setion.6 efinite Integrls 7 rel number suh tht fx) M i i i n sum for ll x in i i i n, then we my define the lower Lf, P P P n ) nd the upper sum Uf, P P P n ) m m i i m m i i m n i n m n i n m i i i n x i x i x nin.6.) M i i i n x i x i x nin,.6.) where x jk is the length of the kth intervl of the prtition P j. We then sy f is integrble on if there exists unique rel number I with the property tht Lf, P P P n ) I Uf, P P P n ).6.) for ll hoies of prtitions P, P,..., P n nd we write I fx, x,..., x n )dx dx dx n,.6.4) or I fx)dx,.6.5) for the definite integrl of f on. We my now generlize the definition of the integrl to more generl regions in the sme mnner s bove. Moreover, our integrbility theorem nd Fubini s theorem, with pproprite hnges, hold s well. When n, we my interpret fx, y, z)dxdydz.6.6) to be the totl mss of if fx, y, z) represents the density of mss t x, y, z), or the totl eletri hrge of if fx, y, z) represents the eletri hrge density t x, y, z). For ny vlue of n we my interpret dx dx dx n.6.7) to be the n-dimensionl volume of. We will not go into further detils, preferring to illustrte with exmples. Exmple Suppose is the losed retngle {x, y, z, t) : x, y, z, t } [, ] [, ] [, ] [, ].
8 efinite Integrls Setion.6 Then x + y + z t )dxdydzdt 6x. x + y + z t )dtdzdydx ) x t + y t + z t t dzdydx x + y + z 8 ) dzdxdy ) x z + y z + z 8z 8x + 8y + ) 8x y + 8y 6x + 6 + 6x ) ) dx dx ) dydx dydx Exmple Let be the region in R bounded by the the three oordinte plnes nd the plne P with eqution z x y see Figure.6.). Suppose we wish to evlute xyzdxdydz. Note tht the side of whih lies in the xy-plne, tht is, the plne z, is tringle with verties t,, ),,, ), nd,, ). Or, stritly in terms of x nd y oordintes, we my desribe this fe s the tringle in the first qudrnt bounded by the line y x see Figure.6.). Hene x vries from to, nd, for eh vlue of x, y vries from to x. Finlly, one we hve fixed vlues for x nd y, z vries from up to P, tht is, to x y. Hene we hve xyzdxdydz x x y x x xyz xyzdzdydx x y dydx xy x y) dydx
Setion.6 efinite Integrls 9 z y, ),, ) z x y y x,, ) y, ) x x,, ) Figure.6. Region bounded by the oordinte plnes nd the plne z x y x 7. xy x y + x y + x y + xy )dydx xy x y + x y + x y ) + xy4 x dx 4 ) x 4 x + 9x x4 + x5 dx ) x 8 x + 9x4 9 8 x5 5 + x6 7 8 9 + 9 8 5 + ) 7 Exmple Let V be the volume of the region in R bounded by the prboloids with equtions z x y nd z x + y 8 see Figure.6.). We will find V by evluting V dxdydz. To set up n iterted integrl, we first note tht the prboloid z x y opens downwrd bout the z-xis nd the prboloid z x + y 8 opens upwrd bout the z xis. The two prboloids interset when x y x + y 8,
efinite Integrls Setion.6 y - x - 5 z -5 Figure.6. Region bounded by z x y nd z x + y 8 tht is, when x + y 9. Now we my desribe the region in the xy-plne desribed by x + y 9 s the set of points x, y) for whih x nd, for every suh fixed x, x y x. Moreover, one we hve fixed x nd y so tht x, y) is inside the irle x + y 9, then x, y, z) is in provided x + y 8 z x y. Hene we hve
Setion.6 efinite Integrls V 8 dxdydz 9 x x y 9 x 9 x 9 x x +y 8 y dzdydx z x x +y 8 dydx 9 x 8 x y )dydx 9 x ) 8y x y y 9 x 6 9 x 4x 9 x 4 9 x ) 9 x 9 x ) dx. 9 x dx 6 4x 4 ) 9 x ) dx Using the substitution x sinθ), we hve dx osθ)dθ, nd so ) dx V 8 8 6 6 54 54 π π π π 9 x ) dx 9 9 sin x)) osθ))dθ θ π π os 4 θ)dθ ) + osθ) dθ + osθ) + os θ))dθ + sinθ) π π + π + os4θ) dθ ) 54π + 7θ π π 8π. + 7 4 sin4θ) π π
efinite Integrls Setion.6 Problems. Evlute eh of the following iterted integrls. ) ) xy dydx b) 4 x y )dxdy d) 4x sinx + y)dydx e x+y dxdy. Evlute the following definite integrls over the given retngles. ) y xy)dxdy, [, ] [, ] b) dxdy, [, ] [, ] x + y) ) ye x dxdy, [, ] [, ] d) dxdy, [, ] [, ] x + y. For eh of the following, evlute the iterted integrls nd sketh the region of integrtion. ) ) y 4 x xy x )dxdy b) 4 x y )dydx d) x x 4 y x + y )dydx xye x y dxdy 4. Find the volume of the region beneth the grph of fx, y) + x + y nd bove the retngle [, ] [, ]. 5. Find the volume of the region beneth the grph of fx, y) 4 x + y nd bove the region {x, y) : x, x y x}. Sketh the region. 6. Evlute xydxdy, where is the region bounded by the x-xis, the y-xis, nd the line y x. 7. Evlute e x dxdy where {x, y) : y, y x }. 8. Find the volume of the region in R desribed by x, y, nd z 4 y 4x. 9. Find the volume of the region in R lying bove the xy-plne nd below the surfe with eqution z 6 x y.. Find the volume of the region in R lying bove the xy-plne nd below the surfe with eqution z 4 x y.. Evlute eh of the following iterted integrls. ) ) 4 x x+y 4 x z )dydxdz b) x yz)dzdydx d) xyzdxdydz x x+y x+y+z wdwdzdydx
Setion.6 efinite Integrls. Find the volume of the region in R bounded by the prboloids with equtions z x y nd z x + y 5.. Evlute xydxdydz, where is the region bounded by the xy-plne, the yzplne, the xz-plne, nd the plne with eqution z 4 x y. 4. If fx, y, z) represents the density of mss t the point x, y, z) of n objet oupying region in R, then fx, y, z)dxdydz is the totl mss of the objet nd the point x, ȳ, z), where nd x m ȳ m z m xfx, y, z)dxdydz, yfx, y, z)dxdydz, zfx, y, z)dxdydz, is lled the enter of mss of the objet. Suppose is the region bounded by the plnes x, y, z, nd z 4 x y. ) Find the totl mss nd enter of mss for n objet oupying the region with mss density given by fx, y, z). b) Find the totl mss nd enter of mss for n objet oupying the region with mss density given by fx, y, z) z. 5. If X nd Y re points hosen t rndom from the intervl [, ], then the probbility tht X, Y ) lies in subset of the unit squre [, ] [, ] is dxdy. ) Find the probbility tht X Y. b) Find the probbility tht X + Y. ) Find the probbility tht XY. 6. If X, Y, nd Z re points hosen t rndom from the intervl [, ], then the probbility tht X, Y, Z) lies in subset of the unit ube [, ] [, ] [, ] is dxdydz. ) Find the probbility tht X Y Z. b) Find the probbility tht X + Y + Z.