# Need help with integral another proof

• Dec 19th 2006, 12:18 PM
Swamifez
Need help with integral another proof
If f(x)>0 for all xElement-ofsymbol [a,b], then
b
Integral sign f>0
a

a) Give an example where f(x)>0 for all xElement-ofsymbol [a,b], and f(x)>0 for some xElement-ofsymbol [a,b], and

b
Integral sign f=0
a

b) Suppose that f(x)>0 for all xElement-of symbol[a,b] and f is continous on at x0 in [a,b] and f(x0)>0. Proove that

b
Integral sign f>0
a
• Dec 19th 2006, 01:57 PM
Plato
I will give you some hints for part b.
Because f is continuous, there is a subinterval of [a,b] on which $\displaystyle f(x) > \frac{{f\left( {x_0 } \right)}}{2}.$
To see how this works note that:
$\displaystyle \varepsilon = \frac{{f\left( {x_0 } \right)}}{2}$
$\displaystyle \exists \delta \left[ {\left| {x - x_0 } \right| < \delta } \right]\quad \Rightarrow \quad \left| {f(x) - f(x_0 )} \right| < \varepsilon .$
• Dec 19th 2006, 02:12 PM
Soroban
Hello, Swamifez!

I'll state the problem in LaTex ... maybe someone can help you.

Quote:

If $\displaystyle f(x) \geq 0\;\;\forall x \in [a,b]$, .then: .$\displaystyle \int^b_af(x)\,dx \:> \:0$

a) Give an example where $\displaystyle f(x) \geq 0\:\;\forall x \in [a,b]$, and $\displaystyle f(x) > 0\;\;\exists x \in [a,b]$,
. . and $\displaystyle \int^b_af(x)\,dx\;=\;0$

b) Suppose that $\displaystyle f(x) \geq 0\;\;\forall x \in [a,b]$ and $\displaystyle f$ is continous at $\displaystyle x_o \in [a,b]$, and $\displaystyle f(x_o) > 0$.
Prove that: .$\displaystyle \int^b_af(x)\,dx\:>\:0$

• Dec 19th 2006, 05:12 PM
ThePerfectHacker
Quote:

If $\displaystyle f(x) \geq 0\;\;\forall x \in [a,b]$, .then: .$\displaystyle \int^b_af(x)\,dx \:> \:0$

a) Give an example where $\displaystyle f(x) \geq 0\:\;\forall x \in [a,b]$, and $\displaystyle f(x) > 0\;\;\exists x \in [a,b]$,
. . and $\displaystyle \int^b_af(x)\,dx\;=\;0$

b) Suppose that $\displaystyle f(x) \geq 0\;\;\forall x \in [a,b]$ and $\displaystyle f$ is continous at $\displaystyle x_o \in [a,b]$, and $\displaystyle f(x_o) > 0$.
Prove that: .$\displaystyle \int^b_af(x)\,dx\:>\:0$

Have you ever heard of the "Limit-Limitation Theorem".
It states that if $\displaystyle f(x)\geq 0$ and $\displaystyle \lim_{x\to c}f(x)$ exists then the limit value is non-negative. (Note $\displaystyle c$ can also be limits at infinity).

Now, the integral is the limit of the Riemann sum. The rest is trivial.

For you other question,
Consider,
$\displaystyle f(x)=\left\{ \begin{array}{c}0, \mbox{ for }-1\leq x<0\\ 1, \mbox{ for }x=0 \\ 0, \mbox{ for }0<x\leq 1\end{array} \right\}$
This function is not always zero.
And because it is continous almost everywhere,
$\displaystyle \int_{-1}^1f(x)dx=0$
• Dec 21st 2006, 03:10 AM
Swamifez
Test preparation... Can someone help me sum up this problem into one juicy proof justifying all the the steps in part a. In part be can someone help me show lower sum is >0 and bigger than the supremum and some it up in one big proof.
• Dec 21st 2006, 02:31 PM
Swamifez
More help with this would be greatly appreciated. Thanks
• Dec 21st 2006, 03:15 PM
Plato
Are you doing an online or a take-home test?
I think that if you are being tested on these concepts then you should have a better grasp of the proof than your postings indicate.
• Dec 21st 2006, 04:20 PM
Swamifez
Plato, neither, the class is officially over and the tests were all inclass, so I couldn't have asked for either. I want to understand each concept, each step by step, so if I take another proof class, it won't be that bad.