Area enclosed by bean curve

Question

I just found this interesting article on Wolfram Mathworld. https://mathworld.wolfram.com/BeanCurve.html

I am interested in the following implicit equation: $$(x^{2}+y^{2})^2=a(x^{3}+y^{3})$$

(The curve can also be expressed by the polar equation $r=a(\sin^{3}{\theta}+\cos^{3}\theta)$.)

The aforementioned article states that the enclosed area of the curve is $A=\dfrac{5}{16}\pi a^{2}$, but does not state how to induce the formula. I want to know how.

Answer 1

Using the polar equation $r=f(\theta)$, the area is given by

$\displaystyle \int_{-\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac 12 f(\theta)^2 d \theta = \frac{5 a^2 \pi}{16}.$