## 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 $A = A(r, a)$?