Una partición
P
{\displaystyle P}
[
a
,
b
]
{\displaystyle [a,b]}
x
:
{
0
,
…
,
n
}
⟶
[
a
,
b
]
{\displaystyle x:\{0,\dots ,n\}\longrightarrow [a,b]}
x
{\displaystyle x}
n
≥
2
{\displaystyle n\geq 2}
x
(
0
)
=
a
{\displaystyle x(0)=a}
x
(
n
)
=
b
{\displaystyle x(n)=b}
P
=
{
x
o
=
a
,
x
1
,
…
,
x
n
−
1
,
x
n
=
b
}
{\displaystyle P=\{x_{o}=a,x_{1},\dots ,x_{n-1},x_{n}=b\}}
[
a
,
b
]
{\displaystyle [a,b]}
P
(
[
a
,
b
]
)
{\displaystyle {\mathcal {P}}([a,b])}
Sea
f
:
[
a
,
b
]
⟶
R
{\displaystyle f:[a,b]\longrightarrow \mathbb {R} }
P
∈
P
(
[
a
,
b
]
)
{\displaystyle P\in {\mathcal {P}}([a,b])}
P
=
{
x
o
=
a
,
x
1
,
…
,
x
n
−
1
,
x
n
=
b
}
{\displaystyle P=\{x_{o}=a,x_{1},\dots ,x_{n-1},x_{n}=b\}}
M
i
=
sup
f
(
[
x
i
−
1
,
x
i
]
)
,
i
=
1
,
…
,
n
{\displaystyle M_{i}=\sup f([x_{i-1},x_{i}]),i=1,\dots ,n}
m
i
=
inf
f
(
[
x
i
−
1
,
x
i
]
)
,
i
=
1
,
…
,
n
{\displaystyle m_{i}=\inf f([x_{i-1},x_{i}]),i=1,\dots ,n}
La suma superior de Riemman de la función
f
{\displaystyle f}
P
{\displaystyle P}
U
(
P
,
f
)
=
∑
i
=
1
n
M
i
Δ
x
i
{\displaystyle U(P,f)=\sum _{i=1}^{n}M_{i}\Delta x_{i}}
Δ
x
i
=
x
i
−
x
i
−
1
,
i
=
1
,
…
,
n
{\displaystyle \Delta x_{i}=x_{i}-x_{i-1},i=1,\dots ,n}
La suma superior de Riemman de la función
f
{\displaystyle f}
P
{\displaystyle P}
L
(
P
,
f
)
=
∑
i
=
1
n
m
i
Δ
x
i
{\displaystyle L(P,f)=\sum _{i=1}^{n}m_{i}\Delta x_{i}}
Δ
x
i
=
x
i
−
x
i
−
1
,
i
=
1
,
…
,
n
{\displaystyle \Delta x_{i}=x_{i}-x_{i-1},i=1,\dots ,n}
Integrales Superiore e Inferior de Riemann
editar
Sea
f
:
[
a
,
b
]
⟶
R
{\displaystyle f:[a,b]\longrightarrow \mathbb {R} }
La integral superior de Riemann de
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
^
f
=
i
n
f
{
U
(
P
,
f
)
:
P
∈
P
(
[
a
,
b
]
)
}
{\displaystyle \int _{a}^{\hat {b}}f=inf\{U(P,f):P\in {\mathcal {P}}([a,b])\}}
La integral inferior de Riemann de
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
f
=
s
u
p
{
L
(
P
,
f
)
:
P
∈
P
(
[
a
,
b
]
)
}
{\displaystyle \int _{a}^{b}f=sup\{L(P,f):P\in {\mathcal {P}}([a,b])\}}
Diremos que
f
:
[
a
,
b
]
⟶
R
{\displaystyle f:[a,b]\longrightarrow \mathbb {R} }
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
f
∈
R
(
[
a
,
b
]
)
{\displaystyle f\in {\mathcal {R}}([a,b])}
i
n
f
{
U
(
P
,
f
)
:
P
∈
P
(
[
a
,
b
]
)
}
=
s
u
p
{
L
(
P
,
f
)
:
P
∈
P
(
[
a
,
b
]
)
}
{\displaystyle inf\{U(P,f):P\in {\mathcal {P}}([a,b])\}=sup\{L(P,f):P\in {\mathcal {P}}([a,b])\}}
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle x:\{0,\dots ,n\}\longrightarrow [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a24a4936374cf92b26204590a39697d2bf69d3)
![{\displaystyle {\mathcal {P}}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b9025311574a9e157a2f0fd9213a0cf388ced7)
![{\displaystyle f:[a,b]\longrightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb09447ab86aaa385a488f810ef8bba56b4b467)
![{\displaystyle P\in {\mathcal {P}}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa95c429a2f8c2f99b7639abc27d7ecc9bf46541)
![{\displaystyle M_{i}=\sup f([x_{i-1},x_{i}]),i=1,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b8c2d7c66acfdd361b1dc8f72fd9900e93485a)
![{\displaystyle m_{i}=\inf f([x_{i-1},x_{i}]),i=1,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248cc16288e596027f8e6837ba12a779a983f1c5)
![{\displaystyle \int _{a}^{\hat {b}}f=inf\{U(P,f):P\in {\mathcal {P}}([a,b])\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd114fe35cbce8f0fee2b7815814341b0b11a218)
![{\displaystyle \int _{a}^{b}f=sup\{L(P,f):P\in {\mathcal {P}}([a,b])\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a408676bf2fcf0376dab4cbe5653c39a28a0e964)
![{\displaystyle f\in {\mathcal {R}}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9b8aaf28b7b398732bd3eb5f700c5b639d8796)
![{\displaystyle inf\{U(P,f):P\in {\mathcal {P}}([a,b])\}=sup\{L(P,f):P\in {\mathcal {P}}([a,b])\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fd7adaf0e163c5d1e66242cbd07bedf04648da)