integration

Estimate ∫f(x)dx by throwing random points into a bounding box. The fraction landing under the curve times the box area approximates the integral.

∫f(x)dx ≈ (points under curve / total points) × box area

MC integration scales to arbitrary dimensions where deterministic methods fail — this is why it powers physics simulations and financial models.

references

Caflisch. "Monte Carlo and quasi-Monte Carlo methods." Acta Numerica, 1998.

Dick et al. "High-dimensional integration: the quasi-Monte Carlo way." Acta Numerica, 2013.