The mean theorem of integral calculus (also called Cauchy's mean theorem ) is an important theorem in analysis . It allows integrals to be estimated without calculating the actual value and provides a simple proof of the fundamental theorem of analysis .
statement
For the geometric interpretation of the mean value theorem for .
![g = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0027652d8e5f694d4aa1c71ce16c9380ce1186)
The Riemann integral is considered here. The statement reads:
Let be a continuous function as well as integrable and either or (i.e. without a change of sign ). Then one exists such that
![{\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ab61178bf5349838758ffe3d96135406ed0245)
![{\ displaystyle g \ colon [a, b] \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38132af5ea7cd916293fe93f29187bd461a5e270)
![g \ ge0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9beeeef79ee73a89cef1e1986fc88d7d248f4e)
![g \ le0](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c06db9f3076b669b40d6407e863a5e2745b967)
![\ xi \ in [a, b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
![\ int \ limits_ {a} ^ {b} {f (x) g (x) dx} = f (\ xi) \ int \ limits_ {a} ^ {b} {g (x) dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9330dcfd6e9d9da0fd10a4c4871ec8a0c38202)
applies. Some authors refer to the above statement as the extended mean theorem and the statement for as the mean theorem or first mean theorem . For you get the important special case:
![g = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0027652d8e5f694d4aa1c71ce16c9380ce1186)
![g = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0027652d8e5f694d4aa1c71ce16c9380ce1186)
-
,
which can be easily interpreted geometrically: The area under the curve between and is equal to the content of a rectangle of medium height .
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
proof
Be on the interval . The other case can be traced back to this by going to.
![g (x) \ ge 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd79defbbb2bed58ca30a66a447cd366a95293c)
![[from]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![-G](https://wikimedia.org/api/rest_v1/media/math/render/svg/698f2bb4b924899b9913617ffb3a3e810fb632aa)
Because of continuity, in takes on a minimum and a maximum according to the principle of minimum and maximum . With and is
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![[from]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![k \ le f (x) \ le K](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbc52dd978c79269bd4b9bca1c68c0753205935)
![g (x) \ ge 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd79defbbb2bed58ca30a66a447cd366a95293c)
-
;
with the monotony and linearity of the Riemann integral
-
.
With therefore applies
![{\ displaystyle I: = \ int \ limits _ {a} ^ {b} {g (x) \, {\ rm {d}} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4352b2cd2eceae949488f98087063eac5f783a16)
-
(1) .
A distinction must now be made between the following cases:
Case I:
. - Then the assertion has the equivalent form
-
;
the right-hand side of this equation is a number, and it has to be shown that one takes this number as its value (2) .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ xi \ in [a, b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
Because is , and (1) has after division by the form
![g (x) \ ge 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd79defbbb2bed58ca30a66a447cd366a95293c)
![I> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/669404ff0ccd51b3c2ad7e789512f97b855e06ca)
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
-
;
from this follows (2) with the intermediate value theorem for continuous functions, q. e. d.
Case II:
. - Then it follows from (1):
-
,
and the assertion takes the form that is valid for everything
![\ xi \ in [a, b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
-
, q. e. d.
Condition on g
The condition that or applies is important. In fact, for functions without this condition , the mean theorem generally does not hold , as the following example shows: For and is
![g \ ge0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9beeeef79ee73a89cef1e1986fc88d7d248f4e)
![-g \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/bded3406867ff047b77deabc55256e8ab62d4094)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![[a, b] = [- 1.1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0a18b1b4c60b123b88b7abe30d73d59016298e)
![f (x) = g (x) = x](https://wikimedia.org/api/rest_v1/media/math/render/svg/34d674f1a6c87ef4ddefbc6fefcd4e7c28e417b4)
-
,
however
-
for everyone .![\ xi \ in [a, b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
Second mean value theorem of integral calculus
Be functions, monotonous and continuous. Then one exists such that
![{\ displaystyle f, g \ colon [a, b] \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07c9cdd885f66cd36e34a6418e28444dfc32ed27)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![\ xi \ in [a, b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
-
.
In the case that it is even continuously differentiable, one can choose. The proof requires partial integration , the fundamental theorem of analysis, and the above theorem.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ xi \ in (a, b)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5abeddc918baa404afdae198bfa6feea783f7e6e)
See also
Web links
literature
-
Otto Forster : Analysis 1. Differential and integral calculus of a variable. 7th edition. Vieweg, Braunschweig 2004, ISBN 3-528-67224-2 .
- Harro Heuser: Textbook of Analysis . Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 .