So Neil, I have run the code on my function and the output came out to be not exactly what I expected. For once, I got multiple column withe the same values in each row. See attached for output. Here is the code I ran

ClearAll[integral3];
integralInY[nx_, ny_, delta_] := 
 integral3[nx, ny, delta] = 
  integral3[nx, ny - delta, delta] + 
   NIntegrate[Func2[x, y], {x, 30, nx}, {y, ny - delta, ny}]
integral3[x_, 5, 1/10] = 0;

And

AbsoluteTiming[
 Export["FileToSaveToFast.csv", 
  Table[integralInY[ii, jj, 1/10], {ii, 30, 139, 1/10}, {jj, 5 + 1/10,
     14, 1/10}]]]

My thought were that I would get the same output of running the following

For[i = 30, i <= 139, i = i + .1, 
 For[j = 5, j <= 14, j = j + .1, 
  PutAppend[NIntegrate[Func2[x, y], {x, 30, i}, {y, 5, j}], 
   "FileToSaveTo.csv"]]]

Any clue what went wrong.

Thanks, Abdullah

Attachments:

Abdullah,

The extension to two dimensions is actually easier than the original, for a fixed x, you integrate in only y. Then you can change your x and integrate in y again:

ClearAll[integral3];
integralInY[nx_, ny_, delta_] := 
 integral3[nx, ny, delta] = 
  integral3[nx, ny - delta, delta] + 
   NIntegrate[func2[x, y], {x, 30, nx}, {y, ny - delta, ny}]
integral3[x_, 5, 1/10] = 0;

To call it and create an array of values:

Table[integralInY[ii, jj, 1/10], {ii, 30, 36, 1/10}, {jj, 5 + 1/10, 
  10, 1/10}]

In this example I go from x = 30 to 35 and y from 5+1/10 to 10 (you MUST start one delta increment larger than 5 based on the definition I chose for the integral. delta is 1/10 in this case. If you want more resolution you need to change the delta in the definition

Neil, thanks for being so responsive. First, I would like you to see the attached file for the output of running your second segment of code. Specifically the following

ClearAll[integral2];
integral2[nx_, ny_, delta_] := 
 integral2[nx, ny, delta] = 
  integral2[nx - delta, ny - delta, delta] + 
   NIntegrate[func2[x, y], {x, 30, nx - delta}, {y, ny - delta, ny}] +
    NIntegrate[func2[x, y], {x, nx - delta, nx}, {y, 0, ny}]
integral2[30, 30, 1] = 0;

And

AbsoluteTiming[
 Export["FileToSaveToFast.csv", 
  Table[integral2[i, i, 1/10], {i, 30, 139, 1/10}]]]

The output came to be the same for both my and your function. I am not sure how were you able to get a valid output, setting delta to 1/10 would always yield a similar output.

Second, regarding implementing your function for different x and y, below are my thoughts

ClearAll[integral2];
integral2[nx_, ny_, delta_] := 
 integral2[nx, ny, delta] = 
  integral2[nx - delta, ny - delta, delta] + 
   NIntegrate[func2[x, y], {x, 30, nx - delta}, {y, ny - delta, ny}] +
    NIntegrate[func2[x, y], {x, nx - delta, nx}, {y, 5, ny - delta}]
integral2[30, 30, 1] = 0;

And

AbsoluteTiming[
 Export["FileToSaveToFast.csv", 
  Table[integral2[i, j, 1/10], {i, 30, 139, 1/10},  {j, 5, 14, 1/10}]]]

Needles to say, this incorrect but I am not really sure for example what this line is for

 integral2[30, 30, 1] = 0;

Regards, Abdullah

Attachments:

Thanks Neil. That is quite the improvement from what I had before. Would you mind however explaining your code a little bit especially how it could be adjusted it so the range of x is between 30 and 139 and the range of y is between 5 and 14 while delta remain same for both, say 0.001. When I run your code, I only get about 10 entries.

Here is what I naively tried doing but the output came out to be a mess

ClearAll[integral2];
integral2[nx_, ny_, delta_] := 
 integral2[nx, ny, delta] = 
  integral2[nx - delta, ny - delta, delta] + 
   NIntegrate[Func2[x, y], {x, 30, nx - delta}, {y, ny - delta, ny}] +
    NIntegrate[Func2[x, y], {x, nx - delta, nx}, {y, 5, ny}]
integral2[30, 30, .001] = 0;
AbsoluteTiming[
 Export["FileToSaveTo.csv", Table[integral2[i, 30, 139], {j, 5, 14}]]]

Thanks,