In your defense, the quadtree method is more easily generalizeable to general figures :)
But indeed I believe there's a very good generalization of the intersection method (the exact one). If you have a general curve and an algebraic description, you can find the intersections roughly using something akin to the quadtree method and then use something like Newton's method to find the precise intersections. It's root finding essentially (basic numerical analysis), you can find very general and good methods on wiki. And then once you have the intersections, you only have to compute some internal polygons and then calculate some integrals. Again, this integrals can be calculated using standard methods like Runge-Kutta. Altogether this gives a very general method with much faster convergence and lower memory (quadratic convergence versus linear convergence for quadtree, provided your boundary satisfies some not too strict conditions).
But indeed I believe there's a very good generalization of the intersection method (the exact one). If you have a general curve and an algebraic description, you can find the intersections roughly using something akin to the quadtree method and then use something like Newton's method to find the precise intersections. It's root finding essentially (basic numerical analysis), you can find very general and good methods on wiki. And then once you have the intersections, you only have to compute some internal polygons and then calculate some integrals. Again, this integrals can be calculated using standard methods like Runge-Kutta. Altogether this gives a very general method with much faster convergence and lower memory (quadratic convergence versus linear convergence for quadtree, provided your boundary satisfies some not too strict conditions).