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I was wondering if this property of integration is held true in the polar coordinates

$$ \int_a^b f(\theta)\,\mathrm{d}\theta = -\int_b^a f(\theta)\,\mathrm{d}\theta $$

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closed as off-topic by Saad, mrtaurho, Vinyl_cape_jawa, José Carlos Santos, Riccardo.Alestra Mar 15 at 15:05

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  • $\begingroup$ Yes, this is valid for any integral and does not depend on a geometric interpretation. $\endgroup$ – Bernard Feb 4 at 8:58
  • $\begingroup$ You can replace $\theta$ with any variable you like, it doesn't make a difference. $\endgroup$ – Qi Zhu Feb 4 at 9:06
  • $\begingroup$ What confused me is that in polar coordinates the limit of integration is in a loop but i think i got it now.. thank you $\endgroup$ – Naifqar Feb 4 at 9:13
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Yes, this is valid for any integral. From the fundamental thereom of calculus: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Corollary $$ \int_a^b f(\theta)\,\mathrm{d}\theta = F(b) - F(a) $$ $$ -\int_b^a f(\theta)\,\mathrm{d}\theta = -(F(a)-F(b))=F(b) - F(a) $$

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This is valid through the FToC. We have a great example with this explanation by Gregory J. Puleo. https://math.stackexchange.com/a/1316543

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