The integral of f(x) = x.^2.*dirac(x) w.r. x from -1 to +1 is zero. That is because the dirac(x) function is considered 0 at all points x except at x = 0, and at x = 0 its value is so large that the integral across the single point x = 0 is just 1. However in the case of your f(x) that one is multiplied by x^2 so that makes the integral zero. If you had a function g(x) = cos(x)*dirac(x), its integral from -1 to +1 would be just cos(0)*1 = 1.
Obviously this cannot be justified in the ordinary numerical sense with finite numbers and only has its basis in what mathematicians call “measure theory”. In particular, you cannot obtain the correct answer using numerical integration.
