Exercise Monte Carlo integration

1. Implement a plain Monte Carlo multi-dimensional integration, fx like
   function plainmc(fun, a, b, N) { // integrates function "fun" over // a rectangular volume defined by // the lower-left point "a" and // the upper-right point "b" // using N pseudo-random points; // returns estimates of the integral // and the integration error var randomx = function(a,b) [a[i]+Math.random()*(b[i]-a[i]) for (i in a)]; var V=1; for(var i in a) V*=b[i]-a[i]; var sum=0,sum2=0; for(var i=0;i<N;i++){ var f=fun(randomx(a,b)); sum+=f; sum2+=f*f} var average =sum/N; var variance=sum2/N-average*average; var Q=V*average; var err=V*Math.sqrt(variance/N); return [Q,err]; }
2. Implement quasi-random sampling.
3. Implement stratified sampling.
4. Calculate ∫₀^π  dx/_π ∫₀^π  dy/_π ∫₀^π  dz/_π [1-cos(x)cos(y)cos(z)]^-1 = Γ(1/4)⁴/(4π³) ≈  1.3932039296856768591842462603255