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    Ok my question is:

    The monte carlo method of integration is surrounding an area/volume with a simpler one, filling this area with random points, then calculating the value of the original area using the fraction of points that fell within in.

    So if you surround a shape with a square, then chuck 50 points in it, if 25 points fell within the original shape it seems reasonable to assume that the shape has an area half that of the square. This is as it is explained here.

    However, as explained here, you do not merely count how many points fall within the shape, but also evaluate the integral at each of these points.

    I know that this works and can use it in practice, but why is the extra step of evaluating the integral at each point necessary? And why is it not necessary in the first case?

    Thanks
 
 
 
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