It's not too much work since most of the fuctions that make the field forces or flux or the shape being analyzed are simple. Just look at the fuctions on top of the integral signs. 5, 5-x, 5-x-y, all simple fuctions to integrate.
Except that's not the functions you integrate. The function you integrate at first is just 1, because it's the integral of dx dy dz, meaning 1 times dx dy dz.
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