I am trying to prove the following:
if $f$ is non negative riemann integrable on $[a,b]$ then there exist $a \leq c \leq d \leq b$ such that $$\int_c^d f(x)dx = 1/2\int_a^b f(x)dx$$
How to prove the theorem about Riemann integrable function?
3
$\begingroup$
real-analysis
riemann-sum
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1hint: use IVT on $\int^x _af(t)dt$. – 2017-02-14
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0what does nonnegative have to do with it? – 2017-02-14
1 Answers
4
Consider $F(x):=\int_a^x f(x)dx$.
$F$ being continuous, the image of interval $[a,b]$ is interval $[0,M]$ with $M:=F(b)=\int_a^b f(x)dx$.
As $F$ takes all the values in $[0,F(b)]$ (by Intermediate Values Theorem), there is, in particular, a value $c$ such that $F(c)=F(b)/2$. (then take $d=b$).