integrale curbilinii de speta a doua
TRANSCRIPT
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8/15/2019 Integrale Curbilinii de Speta a Doua
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UNIVERSITATEA POLITEHNICA BUCURESTI
FACULTATEA DE CHIMIE APLICATA SI STIINTA
MATERIALELOR
PROIECT MATEMATICA
Integrale curbilinii de !eta a II"a
C#nducat#r tiinti$ic Studenti gru!a %%%&A
Pr#$e#r 'ROSU MARTA Iri(ia I#nut"Adrian Ilie R#bert 'abriel
Hanganu Critian
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A!licatii)
%* Sa e calcule+e integrala curbilinie de !eta a d#ua)
2
1
2
11
2
11
2
1
2
1
2
2 2
2
12
12 12
12 2
1
0
2
1
0
e+e=+e+e=I
)t+e+e(=I
∫ dt)t+ee(=I
dt]tee+)e(e∫ +ee[=I
dt=dz
dte=dy
dte=dx
]},o[∈t,t=z,e=y,e=x{=C
∫ xyzdz+xydy+xdx=I
ott
tt
tttttt
t
t
tt
C
,* - Sa e calcule+e c##rd#natele centrului de greutate al $irului (aterial
#(#gen. a/0and denitatea 123. 4* 5 % 6i re!re+entarea !ara(etrica
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5
2
5 5
33
1
0 5 5
33
1
32
32
3
3333
399
1
1
2
5 2
0
5
2
0
44
0
3
2
5 2
0
5
0
2
0
443
2
2 2
0
2
0 0 0
2
2 42 2 42
2 2
a=
tsina
tsina=
tdtcostsinatdtcostsin∫ a=tdtcostsinatsinaa
=y
=tcos
a+tcos
a=
∫ ∫ ∫ tdtsintcosatdtsintcosa=tdtcostsinatcosaa
=x
a=tsin
atsin
a=M
∫ ∫ ∫ ∫ tcostsinatcostsina=tdtcostsina=tdtcostsina=M
tdtcostsina=dttcostsina+tsintcosa=ds
dt)ty′ (+)tx′ (=ds
∫ dsyM
=y
∫ dsxM
=x
w
G
w
G
CG
CG
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π π π π
π
ρ
ρ
7 7
&* Sa e calcule+e lucrul (ecanic al $#rtei F cand !unctul de a!licatie e de!laea+a !edrea!ta C de ecuatii !ara(etrice)3 5 at. 4 5 bt. + 5 ct
2 2 2 z+y+xz
)zk+yj+xi(k=F
∫ nc+!+ac
k=tnc+!+a
c
k=
t
dt
c+!+ac
c+!+ak="
cdt=dz
!dt=dy
adt=dx
dz∫z+y+xz
z+dy
z+y+xz
y+dx
z+y+xz
xk="
∫ ∫ #dz+$dy+%dx=Fdx="
c
C C
2
1
2 2 2 1
2
2 2 2
2 2 2
2 2 2
2 2 2 2 2 2 2 2 2
2