Question

Mean value for integration

Answer 1

The integral mean value theorem (a corollary of the intermediate value theorem) states that a function continuous on an interval takes on its average value somewhere in the interval.

Answer 2

I'm having a problem. I'm going to find the mean value for a integration and that is in it self not that hard we simply have to evaluate for following integral:
\[\large \overline{c}=\int\limits_{0}^{\infty}v ~ f(v) dv\]
My problem is to see reason in why that integral find the mean value.

The problem is that many are familiar with the mean value theorem for integration except that is on a closed interval AKA not working!

Answer 3

The problem is that I'm familiar with the mean value theorem for integration except that it only works on a closed interval so it is not working here.*

Answer 4

you can use the improper integral to define the total value, then half it right?

Answer 5

i think i had some thoughts crossed in that ...

Answer 6

But can you show the outline? How we come to a such result? It just blow my mind we multiply with the variable that is being integrated about.

Answer 7

whats the definite integral that you end up with ... that might help me gather some thoughts

Answer 8

\[mean~value =\lim_{b\to inf}\frac{F(b)-F(a)}{b-a}\]

Answer 9

The definite integral we should end up with is the one I wrote in the question (kinda asking for a proof :) )

Answer 10

\[MVT:\frac{\int_{a}^{b}~f(x)~dx}{b-a}\]
\[\frac{\int_{0}^{b}~v~f(v)~dv}{b}\]
\[\frac{F(b)-\int_{0}^{b}~F(v)~dv}{b}\]
so taking the limit of that as b to infinity
\[\lim_{b\to inf}\frac{F(b)-\int_{0}^{b}~F(v)~dv}{b}\]
would define the value we need to address from 0 to c right?

Answer 11

i dropped a v = b :)

Answer 12

\[\frac{b~F(b)-\int_{0}^{b}~F(v)~dv}{b}\]
\[\lim ~~F(b)-\frac{\int_{0}^{b}~F(v)~dv}{b}\]

Answer 13

thats the best i can make sense of this so far

Answer 14

Right I follow you this long :)

Answer 15

without knowing what F is .... i dont have the wherewithal to take it any further

Answer 16

Well f is going towards 0 when v go towards inf.