Definite Integral on Zero Interval

Theorem

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

Then:

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

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 Darboux sum and upper Darboux 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

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: General Formulas involving Definite Integrals: $15.9$
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Rules and Techniques of Integration: $1.4$