Is possible to extend this answer in order to perform definite integrals in two dimensions step-by-step?
OK, In response to the comment, I did a try and I have by now the following:
showDefiniteIntegral2D[
integrand_, {x_, xMin_, xMax_}, {y_, yMin_, yMax_},
form_: StandardForm] :=
Module[{a, replaceA = "",
antiDerivative = Integrate[integrand, x, y]},
Row[{HoldForm[
Integrate[integrand, {x, xMin, xMax}, {y, yMin, yMax}]], " = ",
Subsuperscript[
DisplayForm[
If[Head[antiDerivative] === Plus,
RowBox[{StyleBox["[", SpanMinSize -> 2],
ToBoxes[antiDerivative, form],
StyleBox["]", SpanMinSize -> 2]}],
RowBox[{ToBoxes[antiDerivative, form],
StyleBox["\[RightBracketingBar]", SpanMinSize -> 2]}]]],
{xMin,yMin},{xMax,yMax}], " = ",
Subtract @@ (antiDerivative /. {x -> {xMax, xMin},
y -> {yMax, yMin}})}]]
I think that the former approach could work,but any suggestion for improvement is welcome.