對(duì)弧長(zhǎng)曲線積分課件

引引例曲線弧L上已知曲線形構(gòu)件在點(diǎn)x,y處的線密度為mxy)求曲線形構(gòu)件的質(zhì)量, M=

,Am(x,y),M,Mn-1yLiMBMMii2Mi

1,

n M?r(xi,hi)DsiyyLMBMi2A,Dsn)l=max(Ds1,,Dsn)

M=limm(xi,hi)Dsi.lfi0iLxOyf(x,yL上有界將L任意分成n個(gè)弧段Ds1,Ds2 ,Dsny并用Dsi表示第i段的弧長(zhǎng)

B在每一弧段n

上任取一點(diǎn)

,hi)

AM

作和f(i,hi)Dsi 令lmax(如果當(dāng)lfi0時(shí)這和的極限總存在則稱此極限為函數(shù)f(x,y)L上對(duì)弧定義記作 f(x,y)ds f(x,y)ds= lfi0f(x,y叫做被積函數(shù)L叫做積分弧段f(x,y)連續(xù)L f(x,y)ds存在.質(zhì)量m=Lm(x, ,m(x,y)為線密度f(wàn)(x,y,z)ds=limfi,hi,zif(x,y)ds

f(x,y)ds

f(x,y)ds1+ L L f(x,y)ds1c1c2為常數(shù)L[c1f(x,y)+c2g(x,y)]ds=c1Lf(x,y)ds+c2Lg(x, L f(x,y)ds=L f(x,y)ds+L f(x,y)d 設(shè)在L上f(x,y)<g(x,y),L

f(x,y)ds￡L g(x,y)ds|L f(x,y)ds|￡L|f(x,y)|ds定理f(x,yL上有定義且連續(xù)L的參數(shù)方程為xj(ty=y

(a￡t￡b)j(t，y(t在[ab上具有一階連續(xù)導(dǎo)數(shù)j2(t+y2(t)0 f(x,y)ds存在,Lf(x,y)dsba2(t)+y2b f(x,y)ds=b

f[f(t),y 2(t)

2 若曲線Ly=y(x)(a￡x￡b

則Lf(xy)dsLxxy=y(x)(a￡x￡bf(x,y)ds=

f[x,y(x)] 1+y2(x)dxb bLxjy)(c￡y￡d則Lf(xy)dsLxjy),yy(c￡y￡df(x,y)ds=

f[j(y), j'2(y)d dx=(3)若曲線G的方程為y=y(tbz=wb

f(x,y,z)dsGf(x,y,z)ds

2(t)

2(t)+w2例計(jì) yds,中L是拋物線y=x2上點(diǎn)O(0,0)與yyy=L解yx20￡x￡11 11 yds=1

2

=

1+4x2

1(55- dI=my2=mdsy2=

(x,y)adsI=Ly2

LL:x=Rcosq,y=Rsinq(-a￡q￡aI(設(shè)線密度為m=1). I R2sin2q(-Rsinq)2+(Rcosq)2=R3

(x2y2z2)ds其中GGxacost,yasintzktt從0到2π)的一段弧解在曲線Gx2+y2+z2=(acost)2+(asint)2+(kt)2=a2+k2tds=(-asint)2+(acost)2+k2dt=a +k2dt

(x2y2z2)ds其中GGxacost,yasintzktt從0到2π)的一段弧

+y2+z2