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Actually, the question about the Rapper doesn't ask how to find the area of a figure bounded by lines with given equations. The question is how to find the area of a square using a multiple integral, so nothing prevents us from setting the coordinate axes parallel to its sides, and we can also place one of the vertices at the origin so that the figure is in the first quadrant, in which case the integrals will be from zero.

And even in the case of sides not parallel to the coordinate axes, an orthogonal transformation of the coordinate system (rotation) does not change the differential of area, and with its help, the axes can be made parallel to the sides.