Graphing Systems of Nonlinear Inequalities

The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure above. For how many ordered pairs of integers $(x, y)$ is the point $(x, y)$ inside or on this curve?

(Hint: For which integer values of $y$ are there points inside or on the curve?)

Source: Math Prize for Girls