The problem is below:
Two snails, Snail A and Snail B, are placed at the origin $(0, 0)$ of a rectangular coordinate system. Snail A starts crawling along a polar curve described by the equation $r = 4 - 3\sin(\theta)$, where $r$ represents the distance from the origin to a point on the curve, and $\theta$ represents the angle in radians. Snail B crawls along a straight line path given by the equation $y = 2x$.
(a) Determine the point(s) where Snail A intersects Snail B.
(b) Calculate the area enclosed by the path of Snail A up to the point(s) of intersection.
I don't understand how to find the intersection... I've tried using $y=2x \implies r\sin(\theta)=2(r\cos\theta)$ but when I simplify, $r = 0$. So how do I get this in terms of $r$? Or how would I convert correctly? Alternatively, I've found $$r = 4 - 3\sin(\theta)$$ can be converted to $$ x^2+y^2=(\frac{3}{4}y+\frac{1}{4}x^2+\frac{1}{4}y^2)^2$$ I'm not sure where to go from here.
The problem is ill posed since it doesn't make sense to start on the cardioid at $r=0$. It's also not clear which half of the cardioid is being traversed. That said, cutting it in half and finding the two areas does make sense.
$r\sin \theta = 2 r \cos \theta \implies \tan \theta=2\implies \cos^2\theta=1/5\implies \sin^2\theta = 4/5\implies \cos 2\theta = -3/5$
$\implies \sin 2\theta = \pm 4/5$
$r=4-3\sin \theta$
$A=\int (1/2)r^2 d\theta = \int (1/2) (16+9\sin ^2 \theta -24\sin\theta) d \theta$
$A=(1/2)[16 \theta + 9 \theta /2-(9/2)\sin \theta \cos \theta+24 \cos \theta]|_0^{2\pi}$
$A=(1/2)[32\pi +9\pi]=41\pi /2$
The area above $y=2x$ contained by the cardioid is
$A_p=(1/2)[16 \theta + 9 \theta /2-(9/2)\sin \theta \cos \theta+24 \cos \theta]|_0^{2\pi}$
$A_p= (1/2)[[16 \arctan 2 + (9/2) \arctan 2-(9/2)(2/5)+24/\sqrt{5}]-[16 \arctan 2 + 8\pi + (9/2)\arctan 2 + 9\pi /4 - (9/2)(2/5)-24\sqrt{5}]]$
$A_p=(1/2)(-41\pi/4+48\sqrt{5})=-41\pi/8+24\sqrt{5}$
With the other area being $A-A_p=205\pi/8-24\sqrt{5}$