Definite Integral on Zero Interval

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Theorem

Let $f$ be a real function which is defined on the closed interval $\Bbb I := \closedint a b$, where $a < b$.


Then:

$\displaystyle \forall c \in \Bbb I: \int_c^c \map f t \rd t = 0$


Proof

Follows directly from the definition of definite integral.

There is only one finite subdivision of $\closedint c c$ and that is $\set c$.

Both the lower sum and upper sum of $\map f x$ on $\closedint c c$ belonging to the finite subdivision $\set c$ are equal to zero.

Hence the result.

$\blacksquare$


Sources