NIntegrate is it possible to evaluate the integral using a user-defined grid?

Question

I am using the NIntegrate function and I would like Mathematica to perform the integration in accordance with a grid I define that is sampled uniformly between the boundaries of integration. Let me illustrate it as follows:

1. We have a function of x to be integrated from 0 to 10
2. We use the values of the integrand at 0, 0.1, 0.2, ..., 9.9, 10.0

Can I use a pre-defined integration rule or should I define the grid points myself?

Answer 1

Reap@NIntegrate[Cos@x, {x, 0, 10}, 
  Method -> {"RiemannRule", "Type" -> "Left", "Points" -> 50}, 
  MaxRecursion -> 0, EvaluationMonitor :> Sow@x]

(*
    {-0.451614, 
    {{0., 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1., 
          1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2., 
          2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3., 
          3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4.,
          4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 5.,
          5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6.,
          6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7., 
          7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9, 8., 
          8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 9.,
          9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9}}}
 *)