calculating the area in polar coördinates

Question

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

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

Thanks in advance

Answer 1

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 $${\rm area}(A)={1\over2}\int_{-\pi}^{\pi}r^2(\theta)\>d\theta=6\pi\ .$$