Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a question which is:

If $f$ is an integrable function on $[a,b]$ and $\int_{a}^{b}{f(x)\,dx}>1$, then there exists a point $c\in(a,b)$ such that $\int_{a}^{c}{f(x)\,dx}=1$.

This seems true to me intuitively. But I can't seem to prove or disprove it. Can someone help me? Thanks :)

share|improve this question

2 Answers 2

up vote 4 down vote accepted

Consider the function $F(s) = \int_a^sf(x)dx$, where $s \in [a,b]$. Clearly, $F(a)=0$, and we know that $F(b)>1$. Since $F(s)$ is continuous, then by Bolzano's theorem there must be some $s \in (a,b)$ such that $F(s) = 1$. That is your $c$.

share|improve this answer
    
Ah thank you :) I was missing the Bolzano part :) –  Jason Feb 11 '12 at 15:22

Yes. The mapping $x\mapsto \int_a^x f(t)\,dt$ is continuous. Apply the intermediate value theorem.

share|improve this answer
    
Sorry, I don't understand. How can you map $x$. You don't know what $f(x)$ is.. –  Jason Feb 11 '12 at 15:16
    
Yes. Edited for that change. –  ncmathsadist Feb 11 '12 at 18:41

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.