Monte Carlo Simulations: How Big Is Your Heart?

Our final problem has no known exact solution. We want to find the area of the shape formed by,

(x^{2} + y^{2} - r^{2})^{3} - a\,x^{2}y^{3} \leq 0

The inequality has two parameters r and a. They are quantities of length, so they take on a nonnegative value. Let’s try out some values and see what these parameters do. I have colored the resulting shapes in red. (Note, the scales are different.)


When r = a (move along the diagonal, starting from bottom-left), we get the shape of a nice heart. When we increase r while holding a fixed, the heart morphs into a circle. On the other hand, when we increase a while holding r fixed, the heart turns into two petals. Well, I see bunny ears.

Clearly, the shape (i.e. area) of our heart depends on the radius r and the ear length a. Can you guess the formula for the area A = A(r, a)?

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