In your case it seems like the function F(x, y) (taken from my example) is = (x^2*y^2)/4. Then you insert the limits like I did in my example to get the same expression as you have on the bottom of your paper.
As I had some integrals that in my computer can not be calculated (not linear, multivariate polynomials)... I thought might be easier to use some scrap paper and hard-code the results inside my code...
Then I can convert the integrals into simple functions of the boundaries.
You can check below what I mean
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