Reversal of Limits of Definite Integral
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Theorem
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be an integrable real function.
Let:
- $\ds \int_a^b \map f x \rd x$
be the definite integral of $f$ over $\closedint a b$.
Let $a \le b$.
Then:
- $\ds \int_a^b \map f x \rd x = -\int_b^a \map f x \rd x$
Proof
\(\ds \int_a^b \map f x \rd x + \int_b^a \map f x \rd x\) | \(=\) | \(\ds \int_a^a \map f x \rd x\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definite Integral on Zero Interval | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_a^b \map f x \rd x\) | \(=\) | \(\ds -\int_b^a \map f x \rd x\) |
Due to the organization of pages at $\mathsf{Pr} \infty \mathsf{fWiki}$, this argument is circular. In particular: Careful with the above. Down at the bottom of Sum of Integrals on Adjacent Intervals for Integrable Functions use is made of this result. There has been a lot of activity on this area of mathematics, and several people have contributed, possibly without having completely followed through all the chains of implications. My bad. Either way this has to be proven independently since the case $a = b$ was not considered in that theorem. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving this issue. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{CircularStructure}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: General Formulas involving Definite Integrals: $15.10$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Rules and Techniques of Integration: $1.3$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.2$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.16$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): definite integral
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): definite integral