Definite integral of a bounded function on a bounded interval depends at least linearly on integration extremes?

Question

I think that the title of the thread summarises all I would like to know about an integral of one variable only. The question may appear silly (maybe it actually is), but it stems from the idea of the raw approximation of an integral provided by calculating the area of a rectangle over the domain of integration.

Answer 1

Let's try to answer your question: Let $f:[-\varepsilon,\varepsilon]\rightarrow\mathbb{R}$ be bounded by a constant $C>0$ and integrable. Then $$|\int_{-\varepsilon}^\varepsilon f(t)\, dt|\leq \int_{-\varepsilon}^\varepsilon |f(t)|\, dt \leq C\int_{-\varepsilon}^{\varepsilon}dt=2C\varepsilon$$ hence the integral is in $O(\varepsilon)$.