Monte Carlo integration method has helped solve many complex problems in a great number of fields such as physics, finance, computer graphics, biology, etc. However, Monte Carlo integration comes with its own set of advantages and disadvantages. Some advantages include simplicity and unbiasedness. Its “simple” because it adapts well to different dimensional integrals. This is because the convergance rate is independent of the integral dimension. That being said, if we fix the number of samples, the quality of approximation does decrease with the number of dimensions, but we can still get a solution. Some disadvantages include slow speed and variance. Monte Carlo integration can have a slow rate of convergence, and, in fact, the convergence rate becomes exponentially worse as the dimension of the integral increases.

Can you explain this a bit more? Isn't the convergence rate independent from the dimension of the integral?

And in terms of "slow speed" what do you mean? Isn't it still faster than most other forms of numerical integration, which is why it's commonly used?