Is $\int_{a}^{b} f(x)\,dx$ equal to $- \int_{b}^{a} f(x)\,dx$?

Asked Jul 30 '19 at 20:20

Active Jul 30 '19 at 20:25

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Is the following equality correct?

$$\int_{a}^{b} f(x)\,dx = - \int_{b}^{a} f(x)\,dx.$$

I think I have already seen it before (I don't remember exactly where), but I can't find nothing about it in the internet. If the answer is yes, can you show the proof?

edited Jul 30 '19 at 20:25

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asked Jul 30 '19 at 20:20

AlfredV

It's correct. You can infer it, I think, from the identity $\displaystyle\int_a^bf(x),dx=\int_a^cf(x),dx+\int_c^bf(x),dx.$ – Adrian Keister Jul 30 '19 at 20:21
That's true. How you prove it depends on how you defined the definite integral and what things you have already proved based on that definition. – Ethan Bolker Jul 30 '19 at 20:22
@PeterForeman You don't need the FTC to prove that identity about definite integrals. – Ethan Bolker Jul 30 '19 at 20:24

Is $\int_{a}^{b} f(x)\,dx$ equal to $- \int_{b}^{a} f(x)\,dx$?

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