When evaluating double integral using polar co-ordinates, does the order of $dr ~ d\theta$ make any difference?


$$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$ $$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 d\theta dr$$

Do the above question yield different answers? If the process for solving the above is different can you please explain in general?

Sorry I have no formatting knowledge, so any light here would be helpful for any future questions.

  • $\begingroup$ Remark about formatting: I fixed your integrals, you can edit your question to see what I did. Math here is generally written in LaTeX, you can learn it from many resources on the Internet. $\endgroup$ – Alfonso Fernandez Mar 20 '13 at 16:31
  • $\begingroup$ In this case, it makes no sense to try to integrate w.r.t. $\theta$ first because the delimiters on $dr$ depend on $\theta$. While it does not make a strict difference, if you want to integrate w.r.t. $\theta$ first, you're going to have to change $\sin\theta$ into something else. $\endgroup$ – Ian Coley Mar 20 '13 at 16:32
  • 1
    $\begingroup$ If you want to change the order of integration, then the limits of integration have to be changed accordingly. See this problem for instance. $\endgroup$ – Mhenni Benghorbal Mar 20 '13 at 16:32

The first integral makes perfect sense; the second one is nonsensical.

I write my integrals with the $d$ first to eliminate any confusion as to what variable goes with what integral:

$$\int_0^{\pi/4} d\theta \: \int_0^{\sin{\theta}} dr \,r^2$$

Evaluate right to left. Clearly, $r$ depends on $\theta$, and integrate over $\theta$. If you wish to switch the order of integration, you must, as Mhenni points out, redefine your integration region:

$$\int_0^{1/\sqrt{2}} dr \, r^2 \int_{\arcsin{r}}^{\pi/4} d\theta$$

| cite | improve this answer | |
  • $\begingroup$ Sooner or later, one should rename the integration tag "Ask Ron Gordon". $\endgroup$ – Julien Mar 20 '13 at 17:09
  • $\begingroup$ @julien: <shakes head> <repeatedly> $\endgroup$ – Ron Gordon Mar 20 '13 at 17:14
  • $\begingroup$ thanks i think i need more work to do in these type of integrations to understand them . $\endgroup$ – adi rohan Mar 20 '13 at 17:23
  • $\begingroup$ One other piece of advice: draw a picture. A picture will make it clear how to define the limits of integration. $\endgroup$ – Ron Gordon Mar 20 '13 at 17:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.