g(x)=f(-x), where $g:[-b,-a] \rightarrow R$ and it is given that $f$ is integrable on $[a,b]$
Need to prove that $\int_{-b}^{-a}f(-x)dx$ is integrable if $\int_{a}^{b}f(x)dx$ is.
Is there a way to do this using the properties of integrals?
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2Isn't this essentially covered by my answer to your other question on the subject? – egreg Jun 11 '16 at 14:23
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You're right, it has, but this was to be proved without Riemann sums/integral. – CyclicalEngineering Jun 11 '16 at 14:24
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Let be $f:[a,b]\to \Bbb{R}$, $a\le b$, integrable. Consider the integral
$$\int_{-b}^{-a}f(-x)dx.$$
Changing the variable, we can take $-x=u$, then $dx=-du$ and the integration limits are $b$ and $a$. So
$$\int_{-b}^{-a}f(-x)dx=\int_b^af(u)(-du)=-\int_{b}^af(u)du=\int_a^bf(u)du.$$
Then $g(x)$ is integrable.
Irddo
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Is that enough to prove it's integrable? Even though all we did is change variables? – CyclicalEngineering Jun 11 '16 at 14:25
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