Question: 2D Grid Interpolation

Hi, 

I'm solving a 2D grid with some finite-diference methods. The result is a surface, i.e f(x,y) = z. Where X and Y and points on a grid. 

I then need to integrate over this grid, i.e

int(f(x,y),[x=0..10, y=0..10]) 

I have tried interpolating the grid. I've used CurveFitting:-ArrayInterpolation() to interpolate points in this 2D space and then integrate over them. 

I'm using a 30x30 grid, but this interpolation scheme takes far too long. The function generated from the ArrayInterpolation creates an interpolation every time a point is evaluated, which I assume is why the integral is very computationally expensive. 

I would like to create a piecewise analytic function from the 2-D grid, perhaps using CurveFitting:-Spline(), however from my understanding this only works for 1D objects? 

Is there any better solution for integrating a 2D numeric grid?

Thanks in advance