Area of a Region Bounded by a Polar Curve, formula with double integral versus single integral using the example of the curve $ x^3+y^3=xy $

Question

My question, when to use which formula to find the area bounded by the curve in polar coordinates? I found such formulas:

$$ \frac{1}{2} \int_\alpha^\beta[f(\theta)]^2 d \theta \text { and } \iint_{\Delta} r d r d \theta $$

Suppose we want to find the area of a figure bounded by a curve $$ x^3+y^3=x y, x \geqslant 0, y \geqslant 0 $$ in polar coordinates it has the equation: $$ r=\frac{\sin \theta \cos \theta}{\cos ^3 \theta+\sin ^3 \theta} {, for } 0 \leqslant \theta \leqslant \frac{\pi}{2} $$ Which formula should I use?

Answer 1

The first formula you listed is just a special case of the second one if $\Delta$ can be parameterized using just one variable in polar coordinates. If you are trying to find the area of

$$\Delta = \left\{(r,\theta) \mid \alpha\le\theta\le\beta, r_1(\theta)\le r<r_2(\theta)\right\},$$

then integrating w.r.t. $r$ yields

$$\iint_\Delta r\,dr\,d\theta = \int_\alpha^\beta \int_{r_1(\theta)}^{r_2(\theta)} r \, dr \, d\theta = \frac12 \int_\alpha^\beta \left(r_2(\theta)^2 - r_1(\theta)^2\right) \, d\theta$$

and you recover the first formula if $f(\theta)=r_2(\theta)$ and $r_1(\theta)=0$ (i.e. $r$ spans the distance from the origin to some polar curve).

For the region bounded by $x^3+y^3=xy \iff r=f(\theta)=\dfrac{\sin\theta\cos\theta}{\sin^3\theta+\cos^3\theta}$, note the self-intersection at $\theta=0$ and $\theta=\dfrac\pi2$. Then the area of $\Delta$ is

$$\int_0^{\tfrac\pi2} \int_0^{f(\theta)} r \, dr \, d\theta = \frac12 \int_0^{\tfrac\pi2} \frac{\sin^2\theta\cos^2\theta}{\left(\sin^3\theta+\cos^3\theta\right)^2} \, d\theta$$

Answer 2

The area is given by \begin{align} Area=&\ \frac{1}{2} \int_0^{\pi/2}r^2(\theta)d \theta \\ = &\ \frac{1}{2} \int_0^{\pi/2} \left(\frac{\sin \theta \cos \theta}{\cos ^3 \theta+\sin ^3 \theta}\right)^2d\theta \overset{t=\tan\theta}= \frac12 \int_0^\infty \frac{t^2}{(1+t^3)^2}dt=\frac16 \end{align}