# Tich Phan Kep

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• CHNG II: TCH PHN BI0: MT S MT BC HAI THNG GP1: TCH PHN KPnh ngha v Cch tnhi bin trong tch phn kpng dng ca tch phn kp2: TCH PHN BI BAnh ngha v Cch tnhi bin trong tch phn bi bang dng ca tch phn bi ba

• 0. Mt s mt bc hai thng gpMt Ellipsoid:2. Cch gi tn mt: Vi phng trnh trn, ta cho x = 0, y = 0, z = 0 ta u nhn c giao tuyn ca mt vi 3 mt ta l cc ng Ellipse. 3. Cch v hnhV 3 giao tuyn ca S vi 3 mt ta Nu c 3 giao tuyn ca 1 mt cong S vi 3 mt ta hoc cc mt song song vi cc mt ta u l ellipse th ta s gi mt S l mt Ellipsoid

• 0. Mt s mt bc hai thng gp

• 0. Mt s mt bc hai thng gp

• 0. Mt s mt bc hai thng gpV mt ellipsoidTrong MatLab, v ellipsoid trn, ta dng lnh ellipsoid(a,b,c)

• 0. Mt s mt bc hai thng gp

• 0. Mt s mt bc hai thng gpII. Mt Paraboloid Elliptic:2. Cch gi tn mt:Vi phng trnh trn, ta cho x = 0, y = 0 th c 2 giao tuyn vi 2 mt ta l 2 ng Parabol v cho z=c, c>0 ta c ng cn li l 1 ng Ellipse. Nu 2 trong 3 giao tuyn vi cc mt ta hoc cc mt song song vi cc mt ta l 2 Parabol, giao tuyn cn li l 1 Ellipse th ta gi mt S l Paraboloid Elliptic

• 0. MT S MT BC HAI THNG GPV ng parabol y2 = z trn mt phng x = 03. V hnh

• 0. MT S MT BC HAI THNG GPV ng ellipse x2+y2 = 1 trn mt phng z = 1

• 0. MT S MT BC HAI THNG GPV mt parabolid x2+y2 = z

• 0. MT S MT BC HAI THNG GPV thm ng parabol x2 = z trn mt phng y = 0

• 0. MT S MT BC HAI THNG GPIII. Mt Tr bc 2:nh ngha mt tr bc 2:Mt tr bc 2 l mt to bi cc ng thng song song vi 1 phng c nh v ta ln 1 ng cong bc 2 c nh. Cc ng thng gi l cc ng sinh ca mt tr, ng cong c nh gi l ng chun ca mt tr.

• Thng thng, ta s ch gp cc mt tr c ng sinh song song vi 1 trong 3 trc ta . 0. MT S MT BC HAI THNG GPMt tr song song vi trc no th phng trnh mt s thiu bin , cn phng trnh bc 2 cha 2 bin cn li l phng trnh ng chun ca mt tr trong mt ta tng ng v ta gi tn mt tr theo tn ca ng chun

• 0. MT S MT BC HAI THNG GPV ng trn x2+y2=1, trn mt z=0Mt tr to bi cc ng thng song song vi Oz v ta ln ng trn trnV d: Mt x2+y2 = 1Phng trnh khng cha z nn n biu din mt tr ng sinh song song vi trc Oz, ng chun l ng trn x2+y2=1 trong mt phng z = 0 v ta gi y l mt tr trn xoay theo tn ca ng chun

• 0. MT S MT BC HAI THNG GPTrong MatLab, v tr trn xoay c th dng lnh cylinder

• 0. MT S MT BC HAI THNG GPV d : Mt z=x2Phng trnh khng cha y nn n biu din mt tr song song vi trc Oy, ng chun l parabol z=x2 trn mt phng y=0 nn ta gi y l mt tr parabolV parabol z=x2 trong mt phng y=0V mt tr c ng sinh song song vi trc Oy, ta ln ng chun l parabol z=x2 trn

• 0. MT S MT BC HAI THNG GPIV. Mt nn bc 2 :Mt nn bc 2 l mt to bi cc ng thng i qua 1 im c nh v ta ln 1 ng cong c nh. Cc ng thng gi l cc ng sinh ca mt nn, ng cong c nh gi l ng chun ca mt nn v im c nh gi l nh ca nnV d: Mt nn x2+y2=z2Ct dc mt nn bi cc mt x=0 hoc y=0 ta c 2 ng thng cng i qua gc ta O, ct ngang bi mt z = c v z = -c , c ty , ta c giao tuyn l 2 ng trn tm ti (0,0,c) v (0,0,-c) bn knh bng c

• 0. MT S MT BC HAI THNG GPV giao tuyn x2+y2=1, z=1V giao tuyn x2=z2, y=0V mt nn x2+y2=z2, ly phn z > 0

• 0. MT S MT BC HAI THNG GPV d: Nhn dng v v mt bc 2 sau z = x2+y2-2xGii: Ta ln lt cho x = 0, y = 0, z = 0 tm 3 giao tuyn ca mt cho vi 3 mt ta x = 0 : z = y2 l phng trnh paraboly = 0 : z = x2-2x l phng trnh parabolz = 0 : 0 = x2+y2-2x l pt ng trn (ellipse)Suy ra mt cho l mt Paraboloid EllipticNHN DNG

• 0. MT S MT BC HAI THNG GPV HNH:V 2 giao tuyn vi 2 mt z = 0, y = 0x=0:.1:2;z=0*x;y=sqrt(2*x-x.^2);plot3(x,y,z)hold ony=-sqrt(2*x-x.^2);plot3(x,y,z)Ta c giao tuyn vi z=0

• 0. MT S MT BC HAI THNG GPhold ony=-2:.2:2;x=1+0*y;z=-1+y.^2;plot3(x,y,z)

• 0. MT S MT BC HAI THNG GPV mt>> [r p]=meshgrid(linspace(0,1,20),linspace(0,2*pi,20));>> mesh(r.*cos(p),r.*sin(p),r.^2)

• 0. MT S MT BC HAI THNG GPV d: Nhn dng v v mt bc 2 sau x2+y2+z2-2z=0Gii: Ta ln lt cho x = 0, y = 0, z = 0 tm 3 giao tuyn ca mt cho vi 3 mt ta NHN DNGx = 0 : y2+z2-2z=0 l pt ng trn (ellipse)y = 0 : x2+z2-2z=0 l pt ng trn (ellipse)z = 0 : 0 = x2+y2l pt ng trn (ellipse)Suy ra mt cho l mt Ellipsoid

• 0. MT S MT BC HAI THNG GP>> theta=linspace(0,pi,20);>> phi=linspace(0,2*pi,20);>> [t p]=meshgrid(theta,phi); >> mesh(sin(t).*cos(p),sin(t).*sin(p),1+cos(t))

• 0. MT S MT BC HAI THNG GPV d: Nhn dng v v mt bc 2 sau y2-z2+2y=0Gii: Pt khng cha x nn n biu din mt tr ng sinh song song vi trc OxNHN DNGTrong mp x = 0 : y2 - z2 + 2y = 0 l pt ng hyperbol tc l ng chun l ng hyperbol.Suy ra mt cho l mt Tr Hyperbol

• >> [x y1] =meshgrid(linspace(-1,1,20),linspace(-4,-2,20));>> z1=sqrt(y1.^2+2*y1);>> mesh(x,y1,z1)>> hold on>> mesh(x,y1,-z1)0. MT S MT BC HAI THNG GPTng t, ta v na cn li ng vi 0
• 0. MT S MT BC HAI THNG GPV d: Nhn dng v v cc mt bc 2 sau:y2-z2+2x2=0x2+2x+2z2-3y=0xy=z21. 2 trong 3 giao tuyn l 2 cp t, giao tuyn th 3 l ellipse nn ta c mt nn ellipse2. 2 trong 3 giao tuyn l 2 parabol, giao tuyn th 3 l ellipse nn ta c mt Paraboloid elliptic3. t x=u+v, y=u-v th ta c ptu2-v2=z2 u2=v2+z2 l pt ca mt nn

• 1: Tch phn kp nh ngha v cch tnhSau y, ta s v hnh khi D l hnh ch nht

• 1: Tch phn kp nh ngha v cch tnhChia min D thnh n phn ty Dij bi cc ng thng song song vi 2 trc Ox, Oy. Ti mi min Dij ly 1 im M(xi,yj) ty Dij

• 1: Tch phn kp nh ngha v cch tnhTh tch cc hnh hp nh vi y di l Dij, trn l phn mt z=f(x,y) s c tnh xp x vi hnh hp ch nht y l Dij, chiu cao l f(xi,yj).

• 1: Tch phn kp nh ngha v cch tnhKhi , vt th ban u c th tch xp x vi tng th tch cc hnh hp ch nht nh xp lin tip nhau

• Chia min D thnh n phn khng dm ln nhau l D1, D2, D3, (cc phn khng c phn chung) tng ng c din tch l S1, S2, S3, Trn mi min Dk ta ly 1 im Mk(xk,yk) ty . Hin nhin tng trn ph thuc vo cch chia min D v cch ly im Mknh ngha tch phn kp : Cho hm f(x,y) xc nh trong min ng, b chn D 1: Tch phn kp nh ngha v cch tnh

• Cho n sao cho max{d(D)} 0 (d(D) l k hiu ng knh ca min D tc l khong cch ln nht gia 2 im bt k thuc D)Nu khi y tng Sn tin n gii hn hu hn S khng ph thuc vo cch chia min D cng nh cch ly im Mk th gii hn S c gi l tch phn kp ca hm f(x,y) trn min D v k hiu l Hm f(x,y) c gi l hm di du tch phn, D l min ly tch phn, ds l yu t din tch. Khi y, ta ni hm f(x,y) kh tch trn min D 1: Tch phn kp nh ngha v cch tnh

• Ch : Nu f(x,y) kh tch trn D th ta c th chia D bi cc ng thng song song vi cc trc ta . Lc Dij s l hnh ch nht vi cc cnh l xi, yj nn Sij = xi. yj v ds c thay bi dxdy. V vy, ta thng dng k hiu 1: Tch phn kp nh ngha v cch tnh

• iu kin kh tch : nh ngha ng cong trn : ng cong C c phng trnh tham s y = y(t), x = x(t) c gi l trn nu cc o hm x(t), y(t) lin tc v khng ng thi bng 0. ng cong C c gi l trn tng khc nu c th chia n thnh hu hn cc cung trn.nh l: Hm lin tc trn 1 min ng, b chn v c bin trn tng khc th kh tch trn min . 1: Tch phn kp nh ngha v cch tnhTnh cht : Cho f(x,y), g(x,y) l cc hm kh tch trn D2.

• Tnh cht 1: Tch phn kp nh ngha v cch tnh3.

• nh l: (V gi tr trung bnh ) 1: Tch phn kp nh ngha v cch tnh

• V d : Cho vt th c gii hn trn bi mt bc hai f(x,y) = 16 x2 2y2, gii hn di bi hnh vung D = [0,2]x[0,2] v gii hn xung quanh bi 4 mt phng x=0, x=2, y=0, y=2. c lng th tch ca vt th trong cc trng hp sau : Chia D thnh 4 phn bng nhau; Chia D thnh 16 phn bng nhau; Chia D thnh 64 phn bng nhau;Chia D thnh 256 phn bng nhau;Tnh th tch vt th1: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnh

• nh l Fubini: (Cch tnh tch phn kp) Cho hm f(x,y) lin tc trn min ng v b chn D 1: Tch phn kp nh ngha v cch tnhab

• 1: Tch phn kp nh ngha v cch tnhx=x1(y)

• Gii cu e) Tnh th tch ca vt th.22=48

• Ta i tch phn ny bng 2 cch Cch 1 : Chiu min D xung trc Ox ta c on [1,4]i theo trc Oy t di ln1: Tch phn kp nh ngha v cch tnh

• Cch 2 : Chiu min D xung trc Oy ta c on [-1,3]A(1,-1)B(1,3)C(4,0)-13i theo trc Ox t tri sang th khng ging nh trn, ta s gp 2 ng BC v AC. Do , ta s chia min D thnh 2 phn D1 v D2D1D2x=-y+41: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnh

• 1: Tch phn kp nh ngha v cch tnhTa cn c th xc nh cn ca tch phn trn m khng cn v hnh nh sau:x = -2, x = 1x2+x-2 = 0Vy ta c -2 x 1, tc l ta ly trong khong 2 nghim ca tam thc f(x) = x2+x-2 nn ta c bt ng thc:x2+x-2 0x 2-x2Tc l, vi x nm trong khong (-2,1) th ng thng y=x nm di ng parabol y = 2-x2. Vy ta cng c

• D1D2D3D4Min D c chia thnh 4 phn 1: Tch phn kp nh ngha v cch tnhV d : Tnh tch phn trong D l min gii hn bi : /4max{|x|,|y|} /2

• Ta cn c th tnh tch phn ny bng cch tnh tch phn trn hnh vung ln tr tch phn trn hnh vung nh1: Tch phn kp nh ngha v cch tnhTng t, ta tnh cho 3 tch phn trn 3 min cn li.

• V d: Tnh tch phn kp D l min gii hn bi -1x1, 0y1D11: Tch phn kp nh ngha v cch tnhD l min gii hn bi -1x1, 0y1V d: Tnh tch phn kp D l min gii hn bi -1x1, 0y1

• Nu ch nhn vo min ly tch phn ny th ta chiu D xung trc no cng nh nhau. 1: Tch phn kp nh ngha v cch tnhTuy nhin, hm di du tch phn s buc ta phi chiu D xung trc Oy

• Chiu min D va v xung trc Ox1: Tch phn kp nh ngha v cch tnhTa v min ly tch phn Ta thy phi chia D thnh 2 phn D1 v D2

• Nhc li v ta cc im M c ta l (x,y) trong ta Descartes. Khi , mi lin h gia x, y v r, l 1: Tch phn kp i bin sang ta cct :

• V d: i cc phng trnh sau sang ta cci sang ta cc m rng bng cch t : Th ta c pt r