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I would like to calculate the area of the region between the $y=x^2$ and $y=x$ using double integrals. The problem is I couldn't find the limits of the variable $r$ in this integral, because the line $y=x$ in polar coordinates is $\theta=\pi/4$ which doesn't depend of $r$.

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  • $\begingroup$ You have a double integral, one integral on $r$, and one on $\theta$. This $\pi/4$ would go on one of your limits for $\theta$. What is your exact region? The region where $y\leq x$ and $y\geq x^2$?. $\endgroup$ – John Doe Apr 27 '17 at 0:10
  • $\begingroup$ @JohnDoe I didn't understand, could you give me the explicit limits of this integral? Thank you $\endgroup$ – user42912 Apr 27 '17 at 0:12
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Assuming that the area you're after is the region enclosed by these two curves:

enter image description here

Then $\theta=\pi/4$ on the upper limit, and on the lower limit it is $0$, since this is the lowest you can go in this quadrant.

Meanwhile for the $r$ limits, given a fixed $\theta$, how far can you go outwards from $0$ until you hit the curve $y=x^2$? This can be solved as follows:

$$y=x^2\implies r\sin\theta=r^2\cos^2\theta\implies r=\tan\theta\sec\theta$$

So $r$ goes up to here. So we have $$\int_0^{\pi/4} \left(\int_0^{\tan\theta\sec\theta}rdr\right)d\theta$$

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  • $\begingroup$ Thank you very much, great answer! $\endgroup$ – user42912 Apr 27 '17 at 14:23
  • $\begingroup$ Where did you plot this graph? $\endgroup$ – user42912 Apr 27 '17 at 14:35
  • $\begingroup$ @user42912 On MATLAB $\endgroup$ – John Doe Apr 27 '17 at 16:05
  • $\begingroup$ I knew Wolfram Alpha and Mathematica as well. I was just curious because your plot seems different from other softwares I know. $\endgroup$ – user42912 Apr 27 '17 at 16:31
  • $\begingroup$ Oh, yes I find MATLAB quite easy for producing graphs $\endgroup$ – John Doe Apr 27 '17 at 16:32

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