(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f be continuous on [a,b] and suppose that [tex] f(x) \geq 0 [/tex] for all x Є [a,b]

Prove that if there exists a point c Є [a,b] such that [tex] f(c) > 0 [/tex], then

[tex] \int_{a}^{b} f > 0 [/tex]

2. Relevant equations

3. The attempt at a solution

Using my books notation,

Suppose P = {x0,x1,....,xn} is a partition of [a,b]. For each i = 1,...,n we let:

[tex] M_i(f) = sup\{f(x) : x \epsilon [x_{i-1},x_i]\} [/tex]

[tex] m_i(f) = inf\{f(x) : x \epsilon [x_{i-1},x_i]\} [/tex]

The upper sum of f with respect to P: U(f,P)

The lower sum of f with respect to P: L(f,P)

The upper integral of f on [a,b]: U(f) = inf{ U(f,P) : P is a partition of [a,b] }

The lower integral of f on [a,b]: L(f) = sup{ L(f,P) : P is a partition of [a,b] }

Suppose there exists a c Є [a,b] such that f(c) > 0 then

there exists a [tex] m_i(f) > 0 [/tex]

thus L(f,P) > 0

but then L(f) > 0, but since f is Riemann integrable:

[tex] 0 < L(f) = U(f) = \int_{a}^{b} f [/tex]

I know I'm kinda leaving out some details but is the outline alright? I have a feeling it's wrong since it was really short...

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# Homework Help: Proof on a property of Riemann integrals

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