# Monotonicity of the riemann integral

1. Apr 13, 2009

### Markjdb

Hi everyone,

For integrable $$f,g:\left[a,b\right]\rightarrow\mathbb{R}$$ with $$f(x)\leq g(x)$$ for all $$x\in\left[a,b\right]$$, it's a basic property of the riemann integral that
$$$\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx$$$

My question is whether the strict version of this inequality holds, i.e. if we have the same hypotheses as above, except with $$f(x)<g(x)$$ for all $$x\in\left[a,b\right]$$, then do we get the following inequality?
$$$\int_a^b f(x)\,dx < \int_a^b g(x)\,dx$$$

This question arose while trying to solve a rather different problem; I feel like it's not true in general, but I haven't yet come up with a counterexample.

2. Apr 13, 2009

### Preno

Yes, you get the strict inequality. The integral of a positive function is obviously positive.

3. Apr 13, 2009

### JG89

If you take a = b then the strict inequality is not true.

4. Apr 14, 2009

### sutupidmath

Like jg89 pointed out, it holds as long as the lower and the upper limits of integration are not the same.

5. Apr 15, 2009