The sum of f(seq(k[i],i=1..m)) over k[i]=rang[i], i=1..m, equals the integral

.

For example,

restart; m := 3;

f := (x,y,z)-> x ^2 + y *z:

rang[1] := 1 .. 6; rang[2] := 5 .. 9; rang[3] := 2 .. 4;

VectorCalculus:-int(f(seq(floor(k[i]), i = 1 .. m)), [seq(k[i], i = 1 .. m)] = Parallelepiped(seq(op(1, rang[j]) .. op(2, rang[j])+1, j = 1 .. m)));

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add(add(add(f(seq(k[i],i=1..m)),k[1]=rang[1]),k[2]=rang[2]),k[3]=rang[3])

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The formula is somewhat more complex in the case of noniteger ends of rangs (range is reserved here). I leave it on your own.