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.


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.

  • $\begingroup$ Yes thank you I know how to find this in cartesian but how I would i find this area using the cartesian form? $\endgroup$ 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 $\endgroup$ 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 ... $\endgroup$ Oct 21 '18 at 16:57
  • $\begingroup$ Check wolfram alpha to solve for y as a function of x $\endgroup$ Oct 21 '18 at 17:02
  • $\begingroup$ But how would i know my limits? $\endgroup$ Oct 21 '18 at 17:04

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