Area from polar coordinates

Question

Find the area bounded by the curve $r=2a\cos(3x)$ which is outside of the circle $r=a$, $a>0$.

I have problem finding the range for the angle. I think $\cos(3x)\ge0$ and $2a\cos(3x)\ge a$ and in this way we can find the range for $x$ and then apply the formula for the area? Am I right? Please help!

Answer 1

HINT: you only need the area outside of the circle, then you need to find all the intervals of $x\in[0,2\pi)$ where $|2a\cos(3x)|\ge a$ and the function is injective (that is, the same part of the curve is not plotted twice in the previously mentioned range of $x$).

Because the area is the same in these intersections then it would be enough to count how many times these areas occur, and multiply this number by the area of one of these intersections.

Graphing the function $f(\theta):=2a\cos(3\theta)(\cos\theta,\sin\theta)$, or from a brief analysis, we can see that it have 3 "petals", thus we only need to find the area between one of these petals and the circle (outside the circle), what is defined by $$A(P)=\frac12\int_{-\pi/6}^{\pi/6} (4a^2\cos^2(3\theta)-a^2)\,\mathrm d\theta=\frac{a^2}3\int_0^{\pi/2}(4\cos^2(\alpha)-1)\,\mathrm d\alpha=a^2\frac{\pi}6$$ Thus the total area is $a^2\pi/2$.