1) con limiti fissi:
a
)
:
F
(
x
)
=
∫
c
d
f
(
x
,
y
)
d
y
;
b
)
:
Φ
(
y
)
=
∫
a
b
f
(
x
,
y
)
d
x
{\displaystyle a):\qquad \ F(x)=\int _{c}^{d}f(x,y)\ dy;\qquad b):\qquad \Phi (y)=\int _{a}^{b}f(x,y)\ dx}
Se
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
R
{\displaystyle \ R}
a
≤
x
≤
b
{\displaystyle a\leq x\leq b}
c
≤
y
≤
d
{\displaystyle c\leq y\leq d}
F
(
x
)
{\displaystyle \ F(x)}
Φ
(
y
)
{\displaystyle \ \Phi (y)}
(
c
,
d
)
{\displaystyle \ (c,d)}
(
a
,
b
)
{\displaystyle \ (a,b)}
d
F
(
x
)
d
x
=
∫
c
d
∂
f
∂
x
d
y
;
d
Φ
(
y
)
d
y
=
∫
a
b
∂
f
∂
y
d
x
;
{\displaystyle \ {dF(x) \over dx}=\int _{c}^{d}{\partial f \over \partial x}dy;\qquad {d\Phi (y) \over dy}=\int _{a}^{b}{\partial f \over \partial y}dx;}
2) con limiti variabili
a
)
F
(
x
)
=
∫
α
(
x
)
β
(
x
)
f
(
x
,
y
)
d
y
;
b
)
Φ
(
y
)
=
∫
γ
(
y
)
δ
(
y
)
f
(
x
,
y
)
d
x
{\displaystyle a)\qquad F(x)=\int _{\alpha (x)}^{\beta (x)}f(x,y)dy;\qquad b)\qquad \Phi (y)=\int _{\gamma (y)}^{\delta (y)}f(x,y)dx}
Se
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
Ω
{\displaystyle \ \Omega }
a
≤
x
≤
b
,
c
≤
y
≤
d
,
{\displaystyle a\leq x\leq b,\ c\leq y\leq d,}
α
(
x
)
,
β
(
x
)
{\displaystyle \ \alpha (x),\ \beta (x)}
(
a
,
b
)
,
{\displaystyle \ (a,b),}
γ
(
y
)
,
δ
(
y
)
{\displaystyle \ \gamma (y),\ \delta (y)}
(
c
,
d
)
,
{\displaystyle \ (c,d),}
F
(
x
)
{\displaystyle \ F(x)}
Φ
(
x
)
{\displaystyle \ \Phi (x)}
(
a
,
b
)
{\displaystyle \ (a,b)}
(
c
,
d
)
.
{\displaystyle \ (c,d).}
d
F
(
x
)
d
x
=
∫
α
(
x
)
β
(
x
)
∂
f
∂
x
d
y
−
d
α
d
x
f
[
x
,
α
(
x
)
]
+
d
β
d
x
f
[
x
,
β
(
x
)
]
,
{\displaystyle {dF(x) \over dx}=\int _{\alpha (x)}^{\beta (x)}{\partial f \over \partial x}dy-{d\alpha \over dx}f[x,\alpha (x)]+{d\beta \over dx}f[x,\beta (x)],}
d
Φ
(
y
)
d
y
=
∫
γ
(
y
)
δ
(
y
)
∂
f
∂
y
d
x
−
d
γ
d
y
f
[
γ
(
y
)
,
y
]
+
d
δ
d
y
f
[
δ
(
y
)
,
y
]
.
{\displaystyle {d\Phi (y) \over dy}=\int _{\gamma (y)}^{\delta (y)}{\partial f \over \partial y}dx-{d\gamma \over dy}f[\gamma (y),y]+{d\delta \over dy}f[\delta (y),y].}
![{\displaystyle {dF(x) \over dx}=\int _{\alpha (x)}^{\beta (x)}{\partial f \over \partial x}dy-{d\alpha \over dx}f[x,\alpha (x)]+{d\beta \over dx}f[x,\beta (x)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c395d16f6abb4dba33f3c88d5cca534cd788620)
![{\displaystyle {d\Phi (y) \over dy}=\int _{\gamma (y)}^{\delta (y)}{\partial f \over \partial y}dx-{d\gamma \over dy}f[\gamma (y),y]+{d\delta \over dy}f[\delta (y),y].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b859c6c081832efdc5bcd054c52f50bb5f27848)