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## 5.1 Monte Carlo integration

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The most common use of Monte Carlo methods is the evaluation of multi-dimensional integrals [24]. Consider first the approximation of an integral obtained from the trapezoidal rule (3.3#eq.1)

Instead of a uniform sampling, imagine an evaluation where the positions are random numbers uniformly distributed in the interval  ; this yields a Monte Carlo integration

 (2)

The convergence rate is of course lower than from a uniform sampling (5.1#eq.1); the strength however appears for integrals in higher dimensions  , where the Monte Carlo error scales as  irrespective of the number of dimensions - instead of the scaling in  that is achieved when using a uniform mesh.