Multivariable integration. Multivariable integration. Iterated integration

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1 Multivrible integrtion Multivrible integrtion Integrtion is ment to nswer the question how muh, depending on the problem nd how we set up the integrl we n be finding how muh volume, how muh surfe re, how muh mss, et. The philosophy of integrtion boils down to breking up the quntity tht we re interested in finding into smll mngeble prts, eh prt of whih is esy to find, nd then dding them up to get the totl. We wnt to do this for multivrible funtions. o we strt by onsidering funtion f(x, y) nd retngulr region tht we wnt to integrte over. In this se we will interpret f(x, y) s height nd we wnt to find volume. Our region tht we will integrte over (i.e., the bse ), whih we will denote by, will onsist of the points x b nd y d. We subdivide up into smll piees so tht for these piees the volume beomes essentilly tht of the volume of tll nd skinny box, nmely f(x k, y k ) A k. The point (x k, y k ) is point inside of the smll subdivision nd A k is the re of the subdivision. o to find n pproximtion for the totl we now dd these ltogether to get Volume f(x k, y k ) A k. This method gives wy to pproximte integrls when we nnot diretly integrte using tools of lulus. Also this is essentilly wy tht omputers do numeril integrtion, omputers just love dding lots of numbers together. To get better pproximtion we tke the limit, where here the limiting proess is refining our subdivision, nottionlly this is P 0 (this nottion is not importnt!) nd we hve Volume = lim f(xk, y k ) A k = P 0 f(x, y) da. This ssumes tht the limit exists, fortuntely for us if we know tht f is ontinuous on then the limit exists, or we sy f is integrble on. In prtiulr for the funtions tht we re interested in the integrl will lwys exist. ine integrtion boils down to dding, nd dding behves niely we get the following properties: k k f(x, y) da ( ) f(x, y) + g(x, y) da = f(x, y) da + g(x, y) da If the region tht we re working on n be split into two prts whih only overlp on their borders, i.e., n be broken into two piees 1 nd 2, then f(x, y) da + f(x, y) da 1 2 If f g on then f(x, y) da Iterted integrtion g(x, y) da Let us ontinue with trying to integrte f(x, y) over the region with x b nd y d. Insted of breking the retngle down into ever more refined smller retngles (s we did lst week), we will find it more onvenient to find the volume by sliing the funtion. For exmple by tking the volume tht we re trying to find nd looking t thin strips where we hold y onstnt. Then the volume of one smll strip is A(y) y where A(y) is the re of the ross setion. Tking limits we n onlude Volume = d A(y) dy. On the other hnd we hve tht our ross setion will look like funtion of x s x rnges between nd b, in prtiulr it will be the funtion f(x, y) (remember tht in our ross setion tht we re holding y fixed). o we hve A(y) = b Putting these together we hve f(x, y) dx. d b f(x, y) dx dy. This is nested integrl or n iterted integrl. When evluting this integrl we lwys work from the inside out, tht is we perform the inside integrl nd then evlute the bounds nd then we go to the next integrl. We ould hve strted this whole onverstion by sliing in different wy, i.e., holding x onstnt. The sme ides rry through nd we n onlude tht b d f(x, y) dy dx. Notie tht we hnged both the order on the bounds nd the order of the d terms. Nottion is importnt. The inside integrl goes with the inside d term nd then we work our wy out step by step. Also, it is useful to keep trk of bounds while doing these integrls, for exmple we ould be more speifi bout

2 the bounds (so tht we re less likely to mke mistke). As n exmple we hve b d f(x, y) dy dx = x=b y=d x= y= f(x, y) dy dx. When f(x, y) = g(x)h(y) we n use properties of onstnts with respet to integrtion to onlude d b ( b g(x)h(y) dx dy = )( d ) g(x) dx h(y) dy We note tht for some funtions (for exmple ones involving bsolute vlue) it is sometimes more onvenient to brek the integrl up into piees. On similr note we n use symmetry in some ses to simplify n integrl. Integrtion beyond retngles While we love our flt things, we will hve to del with things whih re not flt, this inludes hving regions tht re not retngles. We will del with two generl ses. A region is y-simple when it n be desribed by x b nd φ 1 (x) y φ 2 (x). For suh region it is good for us to hold x onstnt nd tke thin vertil strips. Doing this we get b φ2 (x) φ 1 (x) f(x, y) dy dx. A region is x-simple when it n be desribed by y d nd ψ 1 (y) x ψ 2 (y). For suh region it is good for us to hold y onstnt nd tke thin horizontl strips. Doing this we get d ψ2 (y) ψ 1 (y) f(x, y) dx dy. Unlike retngles, hnging the order of integrtion is not s simple s swpping few symbols round. To hnge the order of integrtion we need to hnge the wy tht we desribe our region. We hve the following generl proedure: 1. Write down the urrent bounds. 2. Drw piture, lerly inditing the region tht we re integrting nd (idelly) how we re urrently integrting. 3. elbel ny bounding urves s needed, i.e., hnge y = f(x) to x = f 1 (y). 4. Use the piture to determine how to write down the new bounds. Work from the outside in. 5. Woohoo! Bounds hnged. Chnging the bounds n tke some previously impossible funtion to integrte nd help us to integrte. etting up nd hnging the bounds re some of the most importnt ides from this hpter nd you n expet to be tested on them. Note tht when we hnge bounds it might require us to brek our integrl up into severl piees (i.e., whenever bounding urve hnges). Conversely it might lso llow us to onsolidte severl integrls. Integrtion in polr oordintes We n lso integrte in polr oordintes. The importnt prt bout this proess is how we hop our region up. In Crtesin oordintes the ide is to subdivide the region into smll retngles so tht the re of eh retngle is dx dy or dy dx. In polr oordintes when we subdivide we brek things into smll piees of θ, i.e., dθ, nd smll piees of r, i.e., dr. o tht insted of smll retngles we re looking t piees of irulr wedges. We know how to find the re of wedges (i.e., given irle of rdius r nd entrl ngle of α, the re of the wedge is 1 2 αr2 ) nd so we n determine tht the re of our little piee is r dr dθ. In summry we hve: dy dx in Crtesin oordintes. da = r dr dθ in polr oordintes. We lso need to rewrite the funtion tht we re trying to integrte in terms of r nd θ whih n be done using x = r os θ nd y = r sin θ (nother often used ft is x 2 + y 2 = r 2 ), so we hve f(r os θ, r sin θ) r dr dθ. Where we lso need to desribe our region in terms of r nd θ. While it is possible to hve to integrte r dθ dr this will hppen rrely in prtie. o in generl to desribe region we do the following: 1. Find the bounds for θ, these will either be given or re found by looking for the intersetion of urves. 2. One the bounds for θ re known pik typil θ between the bounds nd look for how r vries, i.e., from the losest urve in the diretion of θ to the frthest urve in the diretion of θ. (If there ever is trnsition between urves we simple brek the integrl into piees.) The best time to use polr oordintes re in the following situtions: We re told to do problem in polr oordintes, or re given urves desribing our region in polr oordintes.

3 Our region n esily be desribed using polr oordintes, prtiulrly true when we hve irles entered t the origin or on the x or y xis. If we hve x 2 + y 2 on the inside of some funtion nd we n t integrte. If we nnot mke progress in Crtesin oordintes. Applitions of integrtion There re severl questions tht we n nswer with integrtion, for exmple, Volume = (height) da where the height is usully mesured s the distne from the surfe z = f(x, y) to the xy-plne. But remembering in this formt hs the dvntge of hndling more generl situtions, i.e., finding the volume between two surfes. In this se we figure out the region we integrte over nd for the height we do top bottom. We n lso use double integrtion to nswer questions bout regions in the plne. We n think of region s orresponding to lmin, i.e., thin plte whih hs vrying thikness or density whih we denote using δ(x, y). The first problem to onsider is the mss. If the density is onstnt the mss is simply found by multiplying the density times the re. When the density vries we n pproximte the mss by the following: subdivide the lmin into piees (smll prts of size A); pproximte the mss of eh piee (i.e., δ(x, y) A); dd the msses up (i.e., δ(x, y) A). Tking the limits of finer subdivision this sum beomes n integrl nd we hve Mss = (density) da = δ(x, y) da. If we were to spin this region round line we ould look t vrious quntities ssoited with this tion. For exmple the (first) moment mesures torsionl effets, speifilly the turning effet provided by this fore. To find the moment of prtile we tke fore times distne to where we rotte round. To find the moment of the lmin we repet the sme ide s bove by finding the moment of eh smll piee of subdivision nd dding. Tking the limits of finer subdivisions this beomes n integrl nd we hve Moment = (distne)(density) da. We let M x denote the moment with respet to spinning round the x-xis nd M y denote the moment with respet to spinning round the y-xis. ine the distne from (x, y) to the x-xis is y nd the distne to the y-xis is x we immeditely hve M x = yδ(x, y) da nd M y = xδ(x, y) da. Given moment we n find the enter of mss (x, y) (the point t whih the lmin blnes) by x = M y M = xδ(x, y) da, δ(x, y) da y = M x M = yδ(x, y) da. δ(x, y) da Alterntively we n think of x s the weighted verge of x where we hve weighted eh x vlue ording to the mss t tht point. When finding the enter of mss it is onvenient to use symmetry. We need symmetry of both the region nd the density funtion. We n lso find the seond moment, or the moment of inerti. This works similr to the moment, the only differene being insted of using (distne) we use (distne) 2. o we hve Inerti = (distne) 2 (density) da. We let I x denote the inerti with respet to spinning round the x-xis, I y denote the inerti with respet to spinning round the y-xis, nd I z the inerti with respet to spinning round the z-xis (i.e., spinning the plne round the origin). ine the distne from (x, y) to the x-xis is y, the distne to the y-xis is x, nd the distne to the origin is x 2 + y 2 we immeditely hve I x = y 2 δ(x, y) da, I y = x 2 δ(x, y) da, I z = (x 2 + y 2 )δ(x, y) da = I y + I x. Chnging gers, let us go bk to the se when we hve z = f(x, y) s surfe. We n then sk the question bout how muh surfe re lies bove prtiulr region. We pproh this s lwys by subdividing into little piees, pproximting eh piee, nd dding bk up. For exmple if we subdivide the region into little retngles of size x by y we n look t wht is hppening t the surfe bove this smll retngle. Beuse we re looking t smll retngle the piee bove it will be lmost flt nd so n be well pproximted by using prllelogrm. In prtiulr the prllelogrm whose two sides re formed by the vetors x, 0, x fx nd 0, y, y f y.

4 elling tht the re of the prllelogrm is then the mgnitude of the ross produt we hve x, 0, x f x 0, y, y fy = x y fx, f }{{} y, 1 = f 2 x + f 2 y + 1 A. = A Tking finer subdivisions we then get better pproximtions to the surfe re nd so we hve urfe re = f 2 x + f 2 y + 1 da. Generlly these integrls re terrible, but we n set them up nd in very few rre ses (i.e., ylinders) we n tully do the integrtion. Triple integrtion Over our short lulus reers we strted with single integrtion, I f(x) dx, where we integrted over n intervl nd hve now lerned double integrtion, f(x, y) da, where we integrted over some region. We n of ourse keep going to higher dimensions nd we now disuss triple integrtion f(x, y, z) dv, where we integrte over some solid in three dimensions. Philosophilly integrtion works the sme. Nmely we tke our solid nd divide it up into little piees. On eh little piee the funtion is essentilly onstnt so tht the ontribution from tht piee will be f(x, y, z) V nd then we dd them ll up, so tht we hve f(x, y, z) dv f(x, y, z) V where the pproximtion gets better s we divide into smller nd smller piees (i.e., s we tke limit). The other nie thing is tht we n gin use mny of the sme tehniques. For exmple to evlute the integrl we n use nested integrtion. There re essentilly six different wys we n hoose to nest, i.e., dx dy dz, dx dz dy,..., dz dy dx; the most nturl for most people is to hoose dz dy dx s this tends to be how regions re desribed, nd the most omplited prt of integrtion omes down to urtely desribing the region we re integrting over. A typil integrl using this order of integrtion will be of the form b φ2 (x) ψ2 (x,y) φ 1 (x) ψ 1 (x,y) f(x, y, z) dz dy dx. In determining the bounds it is usully esiest to work from the outside in, i.e., determine the rnge of vlues for x, given typil x determine the rnge of vlues of y (it is helpful to projet the solid down to the xy-plne for this step), given typil x nd y determine the rnge of vlues of z. Triple integrtion n be used for severl pplitions. For exmple to find the volume of region we hve V = dv (i.e., hop our region into little piees, find the volume of eh little piee, nd dd up to get the totl), though of ourse this quikly beomes double integrl. If our solid hs density funtion, δ(x, y, z) then we n look t severl different quntities relted to the solid inluding the mss, enter of mss, nd moments. We hve the following. Mss: M = (density) dv = δ(x, y, z) dv. Moment: A moment is found by summing mss times distne (relly mss times distne, but tht is nother story), so we hve moment = (distne)(density) dv. The moment of solid is tken with respet to plne nd so the distne is the distne to tht prtiulr plne. We re usully onerned with the xy-, xz-, nd yz-plnes nd these moments re respetively: M xy = zδ(x, y, z) dv M xz = yδ(x, y, z) dv M yz = xδ(x, y, z) dv Center of mss: The enter of mss of solid, (x, y, z), lso orresponds to the weighted verging of the x, y nd z vlues respetively. One we hve the moments nd the mss this point is esy to ompute nd we hve x = M yz M, y = M xz M, z = M xy M. In finding the enter of mss we n often use symmetry to simplify the problem (rell tht we need to hve symmetry both in the solid nd in the density funtion). Integrtion using other oordintes In double integrtion we sw tht we ould integrte either in rtesin oordintes or using polr oordintes. The importnt prt bout swithing ws to mke sure we ounted for everything, i.e.,

5 bounds, rewriting the funtion in terms of new vribles, nd the wy we subdivided re. In prtiulr we hd da = dy dx = r dr dθ. The extr r me from how we now subdivided our region. Chnging oordintes is useful when it helps simplify the desription of the region nd/or mkes the funtion tht we re integrting simpler. We n do similr things in three dimensions. Nmely we hve lerned bout two other oordinte systems nd insted of doing triple integrtion with respet to rtesin oordintes (the dz dy dx tht we hve tlked bout) we n do triple integrtion in one of these oordinte systems. The first oordinte system is ylindril oordintes whih we think of s polr +z, so tht x = r os θ, y = r sin θ, z = z. In this se the importnt thing is tht when we hop up our volume into little piees we hve dv = r dz dr dθ. (Agin we ould integrte in other orders, but this is the most ommon order for desribing our region.) The other thing to remember is to hnge our funtion in terms of r, θ, nd z, in generl we hve f(x, y, z) dv= f(r os θ, r sin θ, z) r dz dr dθ where re pproprite bounds to desribe our region in terms of the vribles for ylindril oordintes. Most of the time we will use ylindril oordintes if our bse is esy to desribe in polr oordinte nd/or our funtion involved x 2 + y 2 terms s these simplify to r 2. The seond oordinte system is spheril oordintes whih hs distne ρ nd diretions φ nd θ, i.e., x = ρ sin φ os θ, y = ρ sin φ sin θ, z = ρ os φ. In this se the importnt thing is tht when we hop up our volume into little piees we hve dv = ρ 2 sin φ dρ dφ dθ. (Agin we ould integrte in other orders, but this is the most ommon order for desribing our region. To remember the orret term just remember the lssi song ho, rho, sine of phi ) The other thing to remember is to hnge our funtion in terms of ρ, φ, nd θ, in generl we hve f(x, y, z) dv = f(ρ sin φ os θ, ρ sin φ sin θ, ρ os φ) ρ 2 sin φ dρ dφ dθ where re pproprite bounds to desribe our region in terms of the vribles for spheril oordintes. Most of the time we will use spheril oordintes if our region is esy to desribe in spheril oordinte nd/or our funtion involved x 2 + y 2 + z 2 terms s these simplify to ρ 2. Jobin In both of the bove ses we hnged our oordinte systems nd we lso hd to tke into ount of how we were dividing up our spe, i.e., in spheril we needed to hve ρ 2 sin φ when we looked t dv. In generl we n look t wht hppens when we hnge our bsis. o insted of integrting with respet to sy x nd y we integrte with respet to u nd v. There re three things tht must be hnged. Nmely: 1. The funtion. 2. The bounds. 3. How our new vribles orrespond to the wy we subdivide, i.e., da or dv. o we strt with orrespondene between vribles u, v nd vribles x, y. In prtiulr we hve wy to tke pir (u, v) to pir (x, y) nd vie vers, i.e., x = x(u, v) u = u(x, y) y = y(u, v) v = v(x, y) To hnge our funtion then we simply use the bove reltionship. imilrly we n use these reltionship to rewrite the urves tht bound our region in terms of our new vribles (tht is write our bounding urves s funtions involving x nd y nd reple x by x(u, v) nd y by y(u, v) nd simplify). This tkes re of two out of the three. To determine the lst prt, nmely wht hppens to da or dv, we look t wht hppens to smll piee in the uv-plne nd wht it orresponds to in the xy-plne. In prtiulr it will orrespond roughly to prllelogrm nd we n use the mgnitude of ross produt to find the re of the prllelogrm. The ross produt is determinnt nd following through we get the orreting term, known s the Jobin x y J(u, v) = u u x y. v v Tking the mgnitude mens in this se tking the bsolute vlue of the determinnt. In prtiulr we hve the following. f(x, y) dx dy = f ( x(u, v), y(u, v) ) J(u, v) du dv

6 In three dimensions something similr hppens, the min differene is now tht the Jobin is funtion of three vribles, i.e., x y z u u u x y z J(u, v, w) =. v x w v y w v z w We will use this method when we re instruted to, but lso when we hve unusul funtions on the inside of the funtion we re integrting nd/or the boundry urves re given by unusul funtions.