Properties of Integrals strict inequalities greater then 0

Question

I have a function $f$ such that $f(x)\geq 0$ for $x \in [0,1]$ then by properties of integrals I know that $\int f(x)dx \geq 0$. suppose that $\int f(x)dx=\frac{22}{7}-\pi$. How can I show that $\frac{22}{7} > \pi$?

Answer 1

If you know $\frac{22}{7}-\pi\geq0$ and you know $\pi$ is irrational, then you also know $\frac{22}{7}-\pi\neq0$, from which the desired result follows (this is not related to integrals).

Answer 2

The assumptions $f(x)\ge 0$ and $$\int f(x)dx=\frac{22}{7}-\pi$$ does not imply$$ \frac{22}{7} > \pi$$ unless you have the condition $f(x)>0$ on some interval of the domain.

Make sure to check the assumpsions again.