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How would you say have some equation in the polar coordinate system as: $$r=3\sin3\theta $$

I know how to find the are of one petal of this shape using polar coordinate integration, but say if I convert this to its Cartesian form this exact polar equation using $ x = r \cos \theta$ , $y= r \sin\theta$, then how would I find the area in when it is in the Cartesian form?

Could anyone give me some clear steps, this will really help me understand better. Thanks in advance.

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As you know we have $$ \sin 3\theta = 3 \sin \theta -4 \sin ^3 \theta $$

Thus your equation in Cartesian coordinates gets a little bit complicated. $$ r = 3 \sin (3\theta ) \implies r^2=3r (3\sin \theta -4\sin ^3 \theta )$$ $$ x^2+y^2=9y -12y \frac {y^2}{x^2+y^2}$$

It is not any easier than polar form.

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  • $\begingroup$ Yes thank you I know how to find this in cartesian but how I would i find this area using the cartesian form? $\endgroup$ – Aurora Borealis Oct 21 '18 at 16:24
  • $\begingroup$ You have to cut it into parts where $y$ is a function of $x$ and use symmetry for other parts. Good luck $\endgroup$ – Mohammad Riazi-Kermani Oct 21 '18 at 16:33
  • $\begingroup$ Could you be nice enough to show me how this is done, I just dont have any examples in the textbook ... $\endgroup$ – Aurora Borealis Oct 21 '18 at 16:57
  • $\begingroup$ Check wolfram alpha to solve for y as a function of x $\endgroup$ – Mohammad Riazi-Kermani Oct 21 '18 at 17:02
  • $\begingroup$ But how would i know my limits? $\endgroup$ – Aurora Borealis Oct 21 '18 at 17:04

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