tt= Map[Graphics[{PointSize[0.02],Point/@Reap[NIntegrate[Boole[x^2+y^2<1/4]/Sqrt[x^2+y^2],{x,-2,2},{y,0,x^2},
Method->{"SymbolicPreprocessing","EvenOddSubdivision"->#[[1]],"SymbolicPiecewiseSubdivision"->#[[2]],
Method->{"GlobalAdaptive",Method->"GaussKronrodRule","SingularityDepth"->\[Infinity]}},PrecisionGoal->3,EvaluationMonitor:>Sow[{x,y}]]]
[[2,1]]},PlotRange->{{-2,2},{0,2}},Axes->True]&,Outer[List,{False,True},{False,True}],{-2}];
Grid[Join[{{"",SpanFromLeft,"SymbolicPiecewiseSubdivision",SpanFromLeft},
{SpanFromAbove,SpanFromBoth,False,True}},{Join[{"EvenOddSubdivision",False},tt[[1]]]},
{Join[{SpanFromAbove,True},grarr[[2]]]}],Dividers->All]

……grarr定义没给。已贴出的部分重现不了你的问题。