y=f(x) [a, b], $abòf(x1)d x1. "xÎ[a, b] $axòf(x1)dx1=I(x) .
xI(x)
1: y=I(x)=axòf(x1)dx1 [a,b]. , Dx0ÞDI(x)0.
: xÎ[a,b], Dx , x+DxÎ[a,b].
DI(x)=xx+Dxòf(x1)dx1
DI=I(x+Dx)-I(x)=xx-Dxòf(x1)dx1- axòf(x1)dx1
|DI|=| xx+Dxòf(x1)dx1|£ xx+Dxò|f(x1)|dx1£MDx
|f(x1)| £M.
2: y=f(x) xÎ[a,b], $ I(x)=f(x).
DI(x)=xx+Dxòf(x1)dx1=f(x) Dx.
xÎ[a,b]
DI(x)/Dx=f(x)
Dx0; xx; f(x) f(x)
lim (DI/Dx)=f(x)
Dx0
2: y=f(x) [a,b], I(x) y=f(x) [a,b].
-.
F(x) F(x) : abòdF(x)=F(b)-F(a). ..: F(x) - f(x), : abòf(x)dx=F(b)-F(a)
11. . .
.
, . òUdV òVdU x1x2òUdV=U*V| x1x2- x1x2òVdU. U , , U , V .
x1x2òf(x)dx z, x . f1(z)dz. x1; x2 z1; z2 , z x1; x2 x. , x1x2òf(x)dx=z1z2òf1(z)dz
:
1.0p/2òxsinxdx=0p/2òxd(-cosx)=-xcosx|0p/2 +0p/2òcosx dx=sinx|0p/2=1
14. , - -. .
y0=y(x0)- - - - , - - - y`=f(x;y); . . . - . .- - - : - - ={(x;y)½½x-x0½<a,½y-y0½<½}. - - f(x;y), =½f(x;y)½, (x;y)Î. , - f(x;y) - - - , - - ½f(x1;y1)-f(x2;y2)½< k½(y1-y2)½, (x1;y1)Î, (x2;y2)Î. - f(x;y) - y - - - , - - . : - f(x;y) - - - -: - f(x;y) - - , - , - - y=y(x), xÎ[x0-h;x0+h], h-min h=min(a;/). : y`=limDx0(y(x+Dx)-y(x))/Dx(y(x+Dx)-y(x))/Dx. - - : (y(x+Dx)-y(x))/Dx=f(x,y(x)), y(x+Dx)= y(x)+Dxf(x,y(x)). Dx , - - -: y(x0+Dx)= y(x0)+Dxf(x0;y(x0)). yk=yk-1+Dxf(xk+yk-1)- .
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13. : , , , , .
( ).
y=f(x). , = =b. , , .
s = lim i=1nåDsi
maxDsi0
sn=i=1nåDsi , .
, a£x£b f(x) f(x) , .
Dyi=f(xi)-f(xi-1).
Dsi=Ö(Dxi)2+(Dyi) 2=Ö1+(Dyi/Dxi) 2 Dxi
: (Dyi/Dxi)=(f(xi)-f(xi-1))/(xi-xi-1)=f(xi),
xi-1<xi< xi. Dsi=Ö1+[f(xi)] 2Dxi.
:
sn=i=1nåÖ1+[f(xi)] 2Dxi
.. f(x) , Ö1+[f(xi)]2 . , :
s= lim i=1nåÖ1+[f(xi)]2Dxi= abòÖ1+[f(xi)]2dx.
maxDsi0
: s= abòÖ1+[f(x)]2dx= abòÖ1+(dy/dx)2dx.
. - p=f(q) (p - , q - ). x=p cosq, y=p sinq.
x=f(q) cosq, y=f(q) sinq. . : s= abòÖ[j(t)]+[y(t)]2dt
dx/dq=f(q) cosq-f(q) sinq
dy/dq=f(q) sinq+f(q) cosq.
(dx/dq) 2+(dy/dq) 2=[f(q)]2+[f(q)]2=p2+p2. : s= q0qòÖp2+p2dq
. , , y=f(x), x=a, x=b. -, . , Q=py2=p[f(x)]2. : V=pabòy2dx=pabò[f(x)]2dx.
. , y=f(x) . a£x£b. y=f(x) [a,b].
, Ds1, Ds2, , Dsn. .
DPi=2p(yi-1+ yi)/2Dsi.
Dsi=Ö(Dxi)2+(Dyi) 2=Ö1+(Dyi/Dxi) 2 Dxi.
Dyi/Dxi=(f(xi)-f(xi-1))/(xi-xi-1)ºf(xi), xi-1<xi< xi; Þ
Dsi=Ö1+f2(xi) Dxi;
DPi=2p(yi-1+ yi)/2Ö1+f2(xi) Dxi.
, :
Pn=pi=1nå[f(xi-1)+f(xi)] Ö1+f2(xi) Dxi.
, .
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P = lim i=1nå[f(xi-1)+f(xi)] Ö1+f2(xi) Dxi=
maxDsi0
lim p i=1nå2f(xi) Ö1+f2(xi) Dxi
maxDsi0
P=2pabòf(x) Ö1+f2(xi) dx.
( ). , x=a, x=b. S=abòf(x)dx.
x=a;y=f1(x);f1(x)£f2(x) x=b;y= f2(x);"xÎ[a,b].
y=c x=y1(y); y=d x=y2(y)
y1=y2 ; yÎ[c,d] S=cdò(y2(y)- y1(y))dy.
. p=p(j), j=a; j=b. jÎ[a; b], a<b. A=A0A1A2 An=B; a=j0<j1<<jn=b;
[jk; jk+1]Î[a; b]; S=åDSk; DSk=1/2p(jk)*p(jk+1)sinjk.
S=1/2åp2(j k)* Djk1/2 abòp(j)dj.
15. .
- - - - -, - - -.
: y=f(x, y).
1. .
f(x, y)=f1(x)*f2(y)~dy/dx=f1(x)*f2(y)~dy/dx=f1(x)dx
y=dy/dx; ∫dy/dx=∫f1(x)dx; φ2(y)=φ1(x)+C
2. .
f(x, y)=f(y/x)
, - - -, y/x=uÞy=xu.(xu)`=f(u), u+xu`=f(u), xu`=f(u)-u, x(du)/dx=f(x)-u. du/(f(u)-u)=dx/x. G(u)=Ln½x½+c. G(y/x)=Ln½x½+c- -.