# What is the definite integral of 0?

$\frac{d}{\mathrm{dx}} \left[C\right] = 0$ where $C$ is any constant.
${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ can be taken as ${\sum}_{{x}_{i} = a}^{b} f \left({x}_{i}\right) \Delta {x}_{i}$, so with $f \left(x\right) = 0$:
${\sum}_{{x}_{i} = a}^{b} 0 \Delta {x}_{i} = 0 {\sum}_{{x}_{i} = a}^{b} \Delta {x}_{i} = 0$
Or, you could say that $f \left(x\right) = 0$, thus it has no area between itself and the x-axis when restricted to a boundary.