Integral of Constant/Definite

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Theorem

Let $c$ be a constant.

$\displaystyle \int_a^b c \ \ \mathrm dx = c \left({b-a}\right)$.


Proof

Let $f_c: \R \to \R$ be the constant function.

By definition:

$\forall x \in \R: f_c \left({x}\right) = c$

Thus:

$\sup \left({f_c}\right) = \inf \left({f_c}\right) = c$

So from Upper and Lower Bounds of Integral‎, we have:

$\displaystyle c \left({b-a}\right) \le \int_a^b c \ \ \mathrm dx \le c \left({b-a}\right)$

Hence the result.

$\blacksquare$


Sources