Integrale lineare
modifica
C
=
{\displaystyle \ C=}
(a, b) dell'asse x ,
f
=
f
(
x
)
,
{\displaystyle \ f=f(x),}
I
c
=
lim
Δ
x
i
→
0
∑
i
f
(
x
i
)
Δ
x
i
=
∫
a
b
f
(
x
)
d
x
=
φ
(
b
)
−
φ
(
a
)
,
{\displaystyle \ I_{c}=\lim _{\Delta x_{i}\to \ 0}\sum _{i}f(x_{i})\Delta x_{i}=\int _{a}^{b}f(x)dx=\varphi (b)-\varphi (a),}
essendo
φ
(
b
)
−
φ
(
a
)
{\displaystyle \varphi (b)-\varphi (a)}
f(x) che si annulla per x=a .
Significato geometrico
modifica
L'integrale considerato rappresenta:
"L'area della regione compresa fra l'asse x , la curva y=f(x) e le ordinate x=a , x=b ".
Se la curva attraversa l'asse x , l'area risulta uguale alla differenza dei due integrali ralativi alle due parti in cui la y=f(x) viene divisa dall'asse x .
Teorema della media
modifica
∫
a
b
f
(
x
)
d
x
=
λ
(
b
−
a
)
,
{\displaystyle \int _{a}^{b}f(x)dx=\lambda (b-a),}
essendo
λ
{\displaystyle \ \lambda }
f(x) in (a,b) .
Se la funzione è continua,
λ
=
f
(
c
)
{\displaystyle \ \lambda =f(c)}
a<c<b.
Formule di integrazione approssimata
modifica
1
)
∫
a
b
d
x
=
h
2
[
(
y
0
+
y
n
)
+
2
(
y
1
+
y
4
+
.
.
.
+
y
n
−
1
)
]
,
{\displaystyle 1)\qquad \int _{a}^{b}dx={h \over 2}[(y_{0}+y_{n})+2(y_{1}+y_{4}+...+y_{n-1})],}
essendo:
h
=
b
−
a
n
{\displaystyle \ h={b-a \over n}}
y
0
,
y
1
,
.
.
.
y
n
{\displaystyle \ y_{0},\ y_{1},...y_{n}}
a,a+h,...a+nh=b . (metodo di Bezout).
2
)
∫
a
b
f
(
x
)
d
x
=
h
3
[
(
y
0
+
y
2
n
)
+
2
(
y
2
+
y
4
+
.
.
.
+
y
2
n
−
2
)
+
4
(
y
1
+
y
3
+
.
.
.
+
y
2
n
−
1
)
]
{\displaystyle 2)\qquad \int _{a}^{b}f(x)dx={h \over 3}[(y_{0}+y_{2n})+2(y_{2}+y_{4}+...+y_{2n-2})+4(y_{1}+y_{3}+...+y_{2n-1})]}
avendo
h
,
y
0
,
.
.
.
y
2
n
{\displaystyle \ h,\ y_{0},...y_{2n}}
Formula per il cambiamento di variabile
modifica
Se si pone:
x
=
φ
(
t
)
,
a
=
φ
(
t
1
)
,
b
=
φ
(
t
2
)
{\displaystyle \ x=\varphi (t),\ a=\varphi (t_{1}),\ b=\varphi (t_{2})}
∫
a
b
f
(
x
)
d
x
=
∫
t
1
t
2
f
[
φ
(
t
)
]
φ
′
(
t
)
d
t
,
{\displaystyle \int _{a}^{b}f(x)\ dx=\int _{t_{1}}^{t_{2}}f[\varphi (t)]\varphi '(t)\ dt\ ,}
quando la funzione
f
(
x
)
{\displaystyle \ f(x)}
(
a
,
b
)
{\displaystyle \ (a,b)}
φ
(
t
)
,
φ
′
(
t
)
{\displaystyle \ \varphi (t),\ \varphi '(t)}
(
t
1
,
t
2
)
{\displaystyle \ (t_{1},\ t_{2})}
φ
′
(
t
)
≠
0
.
{\displaystyle \ \varphi '(t)\neq 0\ .}
Integrale curvilineo
modifica
a) definizioni
:
{
C
=
i
n
t
e
r
v
a
l
l
o
(
a
,
b
)
d
e
l
l
′
a
s
s
e
x
,
f
=
f
(
x
,
y
)
c
o
n
y
=
ϕ
(
x
)
{\displaystyle :\qquad \left\{{\begin{matrix}C=intervallo\ (a,b)\ dell'asse\ x,\\f=f(x,y)\ con\ y=\phi (x)\end{matrix}}\right.}
I
c
=
lim
Δ
x
i
→
0
∑
i
f
[
x
i
,
ϕ
(
x
i
)
]
Δ
x
i
=
∫
γ
f
(
x
,
y
)
d
x
=
{\displaystyle \ I_{c}=\lim _{\Delta x_{i}\to 0}\sum _{i}f[x_{i},\phi (x_{i})]\Delta x_{i}=\int _{\gamma }f(x,y)dx=}
=
∫
a
b
f
[
(
x
,
ϕ
(
x
)
]
d
x
=
∫
a
b
g
(
x
)
d
x
,
{\displaystyle \ =\int _{a}^{b}f[(x,\phi (x)]dx=\int _{a}^{b}g(x)dx,}
essendo
γ
{\displaystyle \ \gamma }
A
B
{\displaystyle \ AB}
A
=
[
a
,
ϕ
(
a
)
]
;
B
=
[
b
,
ϕ
(
b
)
]
.
{\displaystyle \ A=[a,\phi (a)];\qquad B=[b,\phi (b)].}
L'integrale considerato si calcola quindi come un integrale lineare.
Se la curva
γ
{\displaystyle \gamma }
x=x(t), y=y(t) , allora l'intervallo curvilineo diventa:
∫
γ
f
(
x
,
y
)
d
x
=
∫
t
1
t
2
f
[
x
(
t
)
,
y
(
t
)
]
x
′
(
t
)
d
t
.
{\displaystyle \int _{\gamma }^{}f(x,y)dx=\int _{t_{1}}^{t_{2}}f[x(t),y(t)]x'(t)dt.}
b) significato geometrico :
rappresenta l'area della regione di piano limitata dall'assex dalle rette x=a, x=b e dall'arco di curva y=g(x) compreso fra queste rette.
a) definizioni:
{
C
=
ϕ
(
x
)
,
f
=
(
x
,
y
)
c
o
n
y
=
ϕ
(
x
)
,
{\displaystyle \qquad \left\{{\begin{matrix}C=\phi (x),\\f=(x,y)\ con\ y=\phi (x),\end{matrix}}\right.}
con
y
=
ϕ
(
x
)
{\displaystyle \ y=\phi (x)}
I
c
=
lim
Δ
s
i
→
0
∑
i
f
[
x
i
,
ϕ
(
x
i
)
]
Δ
s
i
=
∫
γ
f
(
x
,
y
)
d
s
=
{\displaystyle \ I_{c}=\lim _{\Delta s_{i}\to 0}\sum _{i}f[x_{i},\phi (x_{i})]\Delta s_{i}=\int _{\gamma }{}f(x,y)ds=}
=
∫
a
b
f
[
x
,
ϕ
(
x
)
]
1
+
ϕ
′
2
(
x
)
d
x
,
{\displaystyle \ =\int _{a}^{b}f[x,\phi (x)]{\sqrt {1+{\phi }^{'2}(x)}}\ dx,}
essendo
γ
=
A
B
{\displaystyle \ \gamma =AB}
A
≡
[
a
,
ϕ
(
a
)
]
{\displaystyle \ A\equiv [a,\phi (a)]}
B
≡
[
b
,
ϕ
(
b
)
]
.
{\displaystyle \ B\equiv [b,\phi (b)].}
Se la curva fosse data in forma parametrica, si avrebbe:
∫
γ
f
(
x
,
y
)
d
s
=
∫
a
b
f
[
x
(
t
)
,
y
(
t
)
]
x
′
2
(
t
)
+
y
′
2
(
t
)
d
t
.
{\displaystyle \int _{\gamma }{}f(x,y)ds=\int _{a}^{b}f[x(t),y(t)]{\sqrt {x^{'2}(t)+y^{'2}(t)}}\ dt.}
b) significato geometrico:
rappresenta l'area della regione cilindrica avente per base l'arco
γ
{\displaystyle \ \gamma }
f
[
x
,
φ
(
x
)
]
.
{\displaystyle \ f[x,\varphi (x)]\ .}
Integrale doppio di campo
modifica
C
=
{\displaystyle \ C=}
Ω
{\displaystyle \ \Omega }
x
y
,
{\displaystyle \ xy,}
A
B
C
^
:
y
=
α
(
x
)
{\displaystyle {\widehat {ABC}}:y=\alpha (x)}
A
D
C
^
:
y
=
β
(
x
)
{\displaystyle {\widehat {ADC}}:y=\beta (x)}
C
A
D
^
:
x
=
γ
(
y
)
{\displaystyle {\widehat {CAD}}:x=\gamma (y)}
C
B
D
^
:
x
=
δ
(
y
)
{\displaystyle {\widehat {CBD}}:x=\delta (y)}
f
=
f
(
x
,
y
)
{\displaystyle \ f=f(x,y)}
x
,
y
{\displaystyle \ x,y}
I
c
=
lim
Δ
x
i
→
0
∑
i
f
(
x
i
,
y
i
)
Δ
x
i
Δ
y
i
=
∬
R
g
(
x
,
y
)
d
x
d
y
{\displaystyle \ I_{c}=\lim _{\Delta x{i}\to 0}\sum _{i}f(x_{i},y_{i})\Delta x_{i}\Delta y_{i}=\iint _{R}^{}g(x,y)dxdy}
avendo posto:
{
g
(
x
,
y
)
=
f
(
x
,
y
)
i
n
Ω
,
g
(
x
,
y
)
=
0
i
n
R
e
s
t
e
r
n
a
m
e
n
t
e
a
Ω
.
{\displaystyle {\begin{cases}g(x,y)=f(x,y)\ in\ \Omega ,\\g(x,y)=0\ in\ R\ esternamente\ a\ \Omega \end{cases}}.}
dove
R
{\displaystyle \ R}
Ω
{\displaystyle \ \Omega }
x
=
a
,
x
=
b
,
y
=
c
,
y
=
d
.
{\displaystyle \ x=a,\ x=b,\ y=c,\ y=d\ .}
Calcolo per integrazioni successive
modifica
∬
Ω
f
(
x
,
y
)
d
x
d
y
=
∫
c
d
d
y
∫
γ
(
y
)
δ
(
y
)
f
(
x
,
y
)
d
x
=
∫
a
b
d
x
∫
α
(
x
)
β
(
x
)
f
(
x
,
y
)
d
y
.
{\displaystyle \iint _{\Omega }^{}f(x,y)\ dx\ dy=\int _{c}^{d}dy\int _{\gamma (y)}^{\delta (y)}f(x,y)\ dx=\int _{a}^{b}dx\int _{\alpha (x)}^{\beta (x)}f(x,y)\ dy\ .}
Il valore dell'integrale doppio è indipendente dall'ordine delle due integrazioni successive .
Significato geometrico
modifica
Rappresenta il volume del solido limitato dal piano xy , dalla superficie z=f(x, y) e dalla superficie cilindrica che proietta una parte della superficie z=f(x,y) nella regione Ω ; se la superficie data attraversa il piano xy , il volume risulta uguale alla differenza degli integrali doppi relativi alle due parti in cui la z=f(x,y) viene divisa dal piano xy .
Teorema della media
modifica
∬
Ω
f
(
x
,
y
)
d
x
d
y
=
λ
Ω
¯
,
{\displaystyle \iint _{\Omega }^{}f(x,y)dxdy=\lambda {\bar {\Omega }},}
Ω
¯
=
{\displaystyle \ {\bar {\Omega }}=}
Ω
{\displaystyle \ \Omega }
l
<
λ
<
L
,
{\displaystyle \ l<\lambda <\ L,}
l
{\displaystyle \ l}
L
{\displaystyle \ L}
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
Ω
.
{\displaystyle \ \Omega .}
Se
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
Ω
,
{\displaystyle \ \Omega ,}
λ
=
f
(
x
¯
,
y
¯
)
{\displaystyle \ \lambda =f({\bar {x}},{\bar {y}})}
(
x
¯
,
y
¯
)
{\displaystyle \ ({\bar {x}},{\bar {y}})}
Ω
.
{\displaystyle \ \Omega .}
Teorema di Gauss
modifica
∬
Ω
∂
f
∂
x
d
x
d
y
=
∮
γ
f
(
x
,
y
)
d
y
{\displaystyle \iint _{\Omega }^{}{\partial f \over \partial x}dxdy=\oint _{\gamma }^{}f(x,y)dy}
∬
Ω
∂
f
∂
y
d
x
d
y
=
−
∮
γ
f
(
x
,
y
)
d
x
{\displaystyle \iint _{\Omega }^{}{\partial f \over \partial y}dxdy=-\oint _{\gamma }^{}f(x,y)dx}
essendo
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
Ω
{\displaystyle \ \Omega }
γ
{\displaystyle \ \gamma }
Ω
.
{\displaystyle \ \Omega .}
Formula di Green o di Stokes
modifica
∬
Ω
(
∂
B
∂
x
−
∂
A
∂
y
)
d
x
d
y
=
∮
λ
(
A
d
x
+
B
d
y
)
,
{\displaystyle \iint _{\Omega }^{}({\partial B \over \partial x}-{\partial A \over \partial y})dxdy=\oint _{\lambda }^{}(Adx+Bdy),}
essendo
A
(
x
,
y
)
{\displaystyle \ A(x,y)}
B
(
x
,
y
)
{\displaystyle \ B(x,y)}
Ω
{\displaystyle \ \Omega }
λ
{\displaystyle \ \lambda }
Ω
.
{\displaystyle \ \Omega .}
Le formule esposte servono a trasformare un integrale doppio di campo in un integrale curvilineo o viceversa.
Formula per il cambiamento di variabili
modifica
Se si pone:
{
x
=
ψ
(
u
,
v
)
y
=
φ
(
u
,
v
)
,
{\displaystyle {\begin{cases}x=\psi (u\ ,v)\\y=\varphi (u\ ,v)\ ,\end{cases}}}
essendo le
ψ
{\displaystyle \ \psi }
φ
{\displaystyle \ \varphi }
∧
{\displaystyle \ \wedge }
u
,
v
{\displaystyle \ u,\ v}
J
|
ψ
φ
u
v
|
≠
0
{\displaystyle \ J{\begin{vmatrix}\psi &\varphi \\u&v\end{vmatrix}}\neq 0}
∧
,
{\displaystyle \ \wedge \ ,}
∫
∫
Ω
f
(
x
,
y
)
d
x
d
y
=
∫
∫
∧
f
[
ψ
(
u
,
v
)
,
φ
(
u
,
v
)
]
J
|
ψ
φ
u
v
|
d
u
d
v
,
{\displaystyle \int \int _{\Omega }^{}f(x,\ y)dx\ dy=\int \int _{\wedge }^{}f[\psi (u,v),\varphi (u,v)]J{\begin{vmatrix}\psi &\varphi \\u&v\end{vmatrix}}\ du\ dv\ ,}
dove
∧
{\displaystyle \ \wedge }
(
u
,
v
)
{\displaystyle \ (u,\ v)}
Ω
{\displaystyle \ \Omega }
x
y
.
{\displaystyle \ xy\ .}
Se in particolare si pone:
{
x
=
ρ
cos
θ
y
=
ρ
sin
θ
{\displaystyle {\begin{cases}x=\rho \cos \theta \\y=\rho \sin \theta \end{cases}}}
J
(
φ
,
ψ
ρ
,
θ
)
=
|
cos
θ
−
ρ
cos
θ
sin
θ
ρ
cos
θ
|
=
ρ
{\displaystyle \ J{\begin{pmatrix}\varphi ,&\psi \\\rho ,&\theta \end{pmatrix}}={\begin{vmatrix}\cos \theta &-\rho \cos \theta \\\sin \theta &\rho \cos \theta \end{vmatrix}}=\rho }
e la formula diventa :
∬
Ω
f
(
x
,
y
)
d
x
d
y
=
∬
∧
f
(
ρ
cos
θ
,
ρ
sin
θ
)
ρ
d
ρ
d
θ
.
{\displaystyle \iint _{\Omega }^{}f(x,\ y)\ dx\ dy=\iint _{\wedge }^{}f(\rho \cos \theta ,\ \rho \sin \theta )\ \rho \ d\rho \ d\theta .}
Integrale triplo
modifica
C
=
{\displaystyle \ C=}
V
,
{\displaystyle \ V\ ,}
f
(
x
,
y
,
z
)
{\displaystyle \ f(x,\ y,\ z)}
x
,
y
,
z
{\displaystyle \ x,\ y,\ z}
I
c
=
lim
Δ
x
i
→
0
Δ
y
i
→
0
Δ
x
i
→
0
∑
i
f
(
x
i
,
y
i
,
z
i
)
Δ
x
i
Δ
y
i
Δ
z
i
=
∭
V
f
(
x
,
y
,
z
)
d
x
d
y
d
z
=
{\displaystyle \ I_{c}=\lim _{{{\Delta x_{i}\to 0} \over {\Delta y_{i}\to 0}} \over \Delta x_{i}\to 0}\sum _{i}^{}f(x_{i},\ y_{i},\ z_{i})\Delta x_{i}\Delta y_{i}\Delta z_{i}=\iiint _{V}^{}f(x,\ y,\ z)\ dx\ dy\ dz=}
∭
P
g
(
x
,
y
,
z
)
d
x
d
y
d
z
,
{\displaystyle \iiint _{P}^{}g(x,\ y,\ z)\ dx\ dy\ dz\ ,}
dove
P
{\displaystyle \ P}
V
{\displaystyle \ V}
g
(
x
,
y
,
z
)
=
{
f
(
x
,
y
,
z
)
i
n
V
0
i
n
P
f
u
o
r
i
d
i
V
.
{\displaystyle \ g(x,\ y,\ z)={\begin{cases}f(x,\ y,\ z)\quad \ in\ V\\0\quad \ in\ P\ fuori\ di\ V.\end{cases}}}
Calcolo per integrazioni successive
modifica
∭
V
f
(
x
,
y
,
z
)
d
x
d
y
d
z
=
∫
z
1
z
2
d
z
∫
y
1
(
z
)
y
2
(
z
)
d
y
∫
x
1
(
x
,
y
)
x
2
(
y
,
z
)
f
(
x
,
y
,
z
)
d
x
{\displaystyle \iiint _{V}^{}f(x,\ y,\ z)\ dx\ dy\ dz=\int _{z_{1}}^{z_{2}}dz\int _{y_{1}(z)}^{y_{2}(z)}dy\int _{x_{1}(x,y)}^{x_{2}(y,z)}f(x,\ y,\ z)dx}
essendo :
x
1
(
y
,
z
)
{\displaystyle \ x_{1}(y,\ z)}
x
2
(
y
,
z
)
{\displaystyle \ x_{2}(y,\ z)}
P
1
,
P
2
{\displaystyle \ P_{1},\ P_{2}}
x
{\displaystyle \ x}
V
;
{\displaystyle \ V\ ;}
y
1
(
z
)
{\displaystyle \ y_{1}(z)}
y
2
(
z
)
{\displaystyle \ y_{2}(z)}
y
{\displaystyle \ y}
Q
1
,
Q
2
{\displaystyle \ Q_{1},\ Q_{2}}
x
{\displaystyle \ x}
x
y
{\displaystyle \ xy}
P
1
P
2
;
{\displaystyle \ P_{1}P_{2}\ ;}
z
1
{\displaystyle \ z_{1}}
z
2
{\displaystyle \ z_{2}}
z
{\displaystyle \ z}
x
y
.
{\displaystyle \ xy\ .}
Significato fisico
modifica
rappresenta la massa della regione
V
{\displaystyle \ V}
f
(
x
y
,
z
)
{\displaystyle \ f(x\,\ y,\ z)}
Teorema della media
modifica
∫
∫
∫
V
F
(
x
,
y
,
z
)
d
x
d
y
d
z
=
λ
V
¯
,
{\displaystyle \int _{}^{}\int _{}^{}\int _{V}^{}F(x,y,z)dx\ dy\ dz=\lambda {\bar {V}}\ ,}
essendo
V
¯
{\displaystyle \ {\bar {V}}}
V
{\displaystyle \ V}
λ
{\displaystyle \ \lambda }
Formula per il cambiamento di variabili
modifica
Se si pone
:
{
x
=
φ
(
u
,
v
,
w
)
y
=
ψ
(
u
,
v
,
w
)
z
=
χ
(
u
,
v
,
w
)
{\displaystyle \ :\qquad {\begin{cases}x=\varphi (u,v,w)\\y=\psi (u,v,w)\\z=\chi (u,v,w)\end{cases}}}
essendo le
φ
,
ψ
,
χ
{\displaystyle \ \varphi ,\ \psi ,\ \chi }
V
{\displaystyle \ V}
(
u
,
v
,
w
)
{\displaystyle \ (u,\ v,\ w)}
J
(
φ
,
ψ
,
χ
u
,
v
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{\displaystyle J{\begin{pmatrix}\varphi ,&\psi ,&\chi \\u,&v,&w\end{pmatrix}}\neq 0}
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{\displaystyle \iiint _{V}^{}f(x,y,z)\ dx\ dy\ dz=\iiint _{U}^{}f[\varphi (u,v,w),\psi (u,v,w),\chi (u,v,w)]\cdot {\begin{vmatrix}J{\begin{pmatrix}\varphi ,&\psi ,&\chi \\u,&v,&w\end{pmatrix}}\end{vmatrix}}du\ dv\ dw\ .}
Teorema della divergenza
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{\displaystyle \int _{}^{}\int _{}^{}\int _{V}^{}div.\Phi \ d\tau =\iint _{S}^{}\Phi \times dS}
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{\displaystyle \ div.\Phi ={\partial P \over \partial x}+{\partial Q \over \partial y}+{\partial R \over \partial z}}
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{\displaystyle \ P,Q,R}
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{\displaystyle \ V\ .}
![{\displaystyle 1)\qquad \int _{a}^{b}dx={h \over 2}[(y_{0}+y_{n})+2(y_{1}+y_{4}+...+y_{n-1})],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db48fdedf7cf91a455f0693aed940f48cc55ae96)
![{\displaystyle 2)\qquad \int _{a}^{b}f(x)dx={h \over 3}[(y_{0}+y_{2n})+2(y_{2}+y_{4}+...+y_{2n-2})+4(y_{1}+y_{3}+...+y_{2n-1})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e92dee7f5cd60c7113688cc4ffa207b58b589a7)
![{\displaystyle \int _{a}^{b}f(x)\ dx=\int _{t_{1}}^{t_{2}}f[\varphi (t)]\varphi '(t)\ dt\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1375720d3fc7b49adf5d904bab3633bd729edd)
![{\displaystyle \ I_{c}=\lim _{\Delta x_{i}\to 0}\sum _{i}f[x_{i},\phi (x_{i})]\Delta x_{i}=\int _{\gamma }f(x,y)dx=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e032e011f5ad88f9c65ed0fbd07b4e528baaff08)
![{\displaystyle \ =\int _{a}^{b}f[(x,\phi (x)]dx=\int _{a}^{b}g(x)dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78b40a98e2561efcbb35a74c770770a716915018)
![{\displaystyle \ A=[a,\phi (a)];\qquad B=[b,\phi (b)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948a62e0791e52ceb3f2516ca97ffc35b1c3c3d4)
![{\displaystyle \int _{\gamma }^{}f(x,y)dx=\int _{t_{1}}^{t_{2}}f[x(t),y(t)]x'(t)dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5f73f175a9da0720fef2d933c0ae8c94a6d717)
![{\displaystyle \ I_{c}=\lim _{\Delta s_{i}\to 0}\sum _{i}f[x_{i},\phi (x_{i})]\Delta s_{i}=\int _{\gamma }{}f(x,y)ds=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b335df0f48cb3265590c1823f83de752f1752d15)
![{\displaystyle \ =\int _{a}^{b}f[x,\phi (x)]{\sqrt {1+{\phi }^{'2}(x)}}\ dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9907859629e777a6ef4cd2c52a8b55ddc8b3a14c)
![{\displaystyle \ A\equiv [a,\phi (a)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2441f8846258d30416b8a3a52f165ece8734aa)
![{\displaystyle \ B\equiv [b,\phi (b)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83fbe178ae8e5a964f15ee4196e4f871ce66485a)
![{\displaystyle \int _{\gamma }{}f(x,y)ds=\int _{a}^{b}f[x(t),y(t)]{\sqrt {x^{'2}(t)+y^{'2}(t)}}\ dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1176e1293afef8d039c1e46cc1efe1e28a5301)
![{\displaystyle \ f[x,\varphi (x)]\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e00d5313c850ee2521b341dfe0be5f87d94e5ed2)
![{\displaystyle \int \int _{\Omega }^{}f(x,\ y)dx\ dy=\int \int _{\wedge }^{}f[\psi (u,v),\varphi (u,v)]J{\begin{vmatrix}\psi &\varphi \\u&v\end{vmatrix}}\ du\ dv\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4789d6e0dfea9d3e3d7174d45ce31e0a3da964b0)
![{\displaystyle \iiint _{V}^{}f(x,y,z)\ dx\ dy\ dz=\iiint _{U}^{}f[\varphi (u,v,w),\psi (u,v,w),\chi (u,v,w)]\cdot {\begin{vmatrix}J{\begin{pmatrix}\varphi ,&\psi ,&\chi \\u,&v,&w\end{pmatrix}}\end{vmatrix}}du\ dv\ dw\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcb15bd0d36e672ed63f014ee8577d3b307a00e)
