Calculate the area inside the loop

Question

The question is - how to calculate the area inside the loop $(x+y)^3=axy$, using double integral?

In this case parameter $a$ is positive. It's easy to imagine how it looks like, depending on $a$ value. For example $a=1$:

or $a=100$

No idea how to do that.

Answer 1

HINT:

In polar coordinates, the loop is described by

$$r(\theta)=\frac{a\sin(\theta)\cos(\theta)}{(\cos(\theta)+\sin(\theta))^3}$$

$\theta \in [0,\pi/2]$. The area, $A$, is then given by

$$A=\int_0^{\pi/2}\int_0^{r(\theta)}r\,dr\,d\theta \tag 1$$

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SPOILER ALERT: Scroll over the highlighted area to reveal the solution

From $(1)$, we have $$\begin{align}A&=\frac{a^2}2\int_0^{\pi/2} \frac{\sin^2(\theta)\cos^2(\theta)}{(\cos(\theta)+\sin(\theta))^6}\,d\theta\\\\&=\frac{a^2}2 \int_0^{\pi/2}\frac{\tan^2(\theta)\sec^2(\theta)}{(1+\tan(\theta))^6}\,d\theta\\\\&=\frac{a^2}2\int_0^\infty \frac{x^2}{(1+x)^6}\,dx\\\\&=\frac{a^2}{60}\end{align}$$