Generalization of Mean Value Theorem for Integrals Needed

[Only a Mirage]

Hi all,

I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't.


Is the following true?

If [itex] F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n} [/itex] is a continuous function

and [itex]x: I \subset \mathbb{R} \rightarrow V \subset \mathbb{R}^{n}[/itex] is a continuous function

then [itex]\exists t^* \in [t_1,t_2][/itex] such that

[itex]\int_{t_1}^{t_2} F(x(t),t)\,dt = F(x(t^*),t^*)(t_2-t_1)[/itex]

I can see that it holds for each of the component functions of ##F##, but I'm not sure about the whole thing.

 

[Only a Mirage]

I'm actually fairly certain now that my conjecture is false...

 

[pasmith]

Hi all,

I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't.


Is the following true?

If [itex] F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n} [/itex] is a continuous function

and [itex]x: I \subset \mathbb{R} \rightarrow V \subset \mathbb{R}^{n}[/itex] is a continuous function

then [itex]\exists t^* \in [t_1,t_2][/itex] such that

[itex]\int_{t_1}^{t_2} F(x(t),t)\,dt = F(x(t^*),t^*)(t_2-t_1)[/itex]

I can see that it holds for each of the component functions of ##F##, but I'm not sure about the whole thing.

[itex]g(t) = F(x(t),t)[/itex] is a function from [itex][t_1,t_2] \subset \mathbb{R}[/itex] to [itex]\mathbb{R}^n[/itex], so by definition one integrates it component by component with respect to the standard basis.

For each component [itex]g_k[/itex], there exists [itex]t^*_k \in [t_1,t_2][/itex] such that [itex](t_2 - t_1) g_k(t^*_k) = \int_{t_1}^{t_2} g_k(t)\,\mathrm{d}t[/itex] by the mean value theorem applied to [itex]G_k(t) = \int_{t_1}^t g_k(s)\,\mathrm{d}s[/itex].

The value of [itex]t[/itex] where [itex]g_k[/itex] attains its average value is not necessarily unique, so for [itex]g[/itex] to attain its average there must be at least one [itex]t[/itex] where every component attains its average, which is not necessarily the case.

For example, consider [itex]g : [0,1] \to \mathbb{R}^2 : t \mapsto (t,t)[/itex]. Then
[tex]
\int_0^1 g(t)\,\mathrm{d}t = \int_0^1 (t,t^2)\,\mathrm{d}t = (\frac12, \frac13)
[/tex]
Since each component is strictly increasing on [itex][0,1][/itex], that component attains its average at exactly one point, and we have [itex]t_1^{*} = \frac 12[/itex] and [itex]t_2^{*} = \frac{1}{\sqrt{3}}[/itex]. These are not equal, so [itex]g[/itex] does not attain its average on [itex][0,1][/itex].

One can obtain [itex]g[/itex] by taking [itex]F(x,y,z) = (x,y^2)[/itex] and [itex](x(t),y(t)) = (t,t)[/itex].

 

[Only a Mirage]

That makes sense. Thanks a lot for the help.

 

[blue_raver22]

Dear researcher,

Thank you for your inquiry about a generalization of the mean value theorem for integrals. This is an interesting and challenging topic, and it is important to carefully consider any conjectures before attempting to prove them.

From the information provided, it seems that you are interested in the possibility that the mean value theorem for derivatives can be extended to integrals in higher dimensions. This is a common question, and there are indeed several generalizations of the mean value theorem for integrals.

One such generalization is the Mean Value Theorem for Integrals in Higher Dimensions, which states that if F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n} is a continuous function and x: I \subset \mathbb{R} \rightarrow V \subset \mathbb{R}^{n} is a continuous function, then there exists a point (x^*,t^*) \in U \times I such that

\int_{t_1}^{t_2} F(x(t),t)\,dt = F(x^*,t^*)(t_2-t_1)

This generalization is often used in multivariable calculus and has been proven to hold under certain conditions.

Another generalization is the Mean Value Theorem for Integrals on Curves, which states that if x: I \subset \mathbb{R} \rightarrow V \subset \mathbb{R}^{n} is a continuous function and f: V \rightarrow \mathbb{R} is a continuous function, then there exists a point t^* \in I such that

\int_{t_1}^{t_2} f(x(t))\,|x'(t)|\,dt = f(x(t^*))(t_2-t_1)

This generalization is commonly used in the study of curve integrals and has also been proven to hold under certain conditions.

In conclusion, there are indeed generalizations of the mean value theorem for integrals, but it is important to carefully consider the conditions under which they hold. I hope this information helps in your research. Best of luck in your endeavors.

Sincerely,