# calculating the area in polar coördinates

I have difficulties calculating the area and setting the right boundaries of the following polar coördinates:

$$r=2(1+cos(\theta) )$$

• Is $a$, the polar angle (generally referred as) $\theta$? – Anurag A Jun 3 at 17:38
• yes it is, i did not know how to type it – Wouter Lommerse Jun 3 at 17:39
• Since you haven't shown the detailed steps of your work we can only guess where you made your error. There are various ways you could make a mistake by a factor of $2$ in a problem like this. – David K Jun 3 at 17:51
• Looks like you only squared part of your r @wouterlommerse – randomgirl Jun 3 at 19:51
• It generally works out better (and I think is probably easier for you too) to edit the question in order to add information rather than putting a lot of formulas in comments. But it looks like the error has been identified now. – David K Jun 3 at 22:19

The function $$\theta\mapsto r(\theta):=2(1+\cos\theta)$$ does not define an area per se. Now this function is $$2\pi$$-periodic, and graphing the curve $$\gamma:\quad\theta\mapsto\bigl(x(\theta),y(\theta)\bigr)=r(\theta)\,(\cos\theta,\sin\theta)\qquad(-\pi\leq\theta\leq\pi)$$ we obtain a "loop with an indent" enclosing a certain shape $$A$$, whereby $$\gamma$$ is astroidal with respect to the origin. The area of $$A$$ then can be calculated with the formula $${\rm area}(A)={1\over2}\int_{-\pi}^{\pi}r^2(\theta)\>d\theta=6\pi\ .$$