CDF[MultinormalDistribution[{0,0},({{1,37/40},{37/40,1}})],{0,0.2}]

0.446357

NIntegrate[PDF[MultinormalDistribution[{0,0},({{1,37/40},{37/40,1}})],{x, y}],{x,-[Infinity],0},{y,-[Infinity],0.2}]

0.470073

dist = MultinormalDistribution[{0, 0}, ({{1, 37/40}, {37/40, 1}})];
CDF[dist, {0, 0.2`3}]
Precision[0.2]
$MachinePrecision
CDF[dist, {0, 0.2}]
(*  0.47  *)
(*  MachinePrecision  *)
(*  15.9546  *)
(*  0.446357  *)

NMinimize[{CDF[dist, {0, y}], 0.0 <= y <= 0.4}, y]
NMaximize[{CDF[dist, {0, y}], 0.0 <= y <= 0.4}, y]
(*  {0.437977, {y -> 0.}}  *)
(*  {0.465287, {y -> 0.4}}  *)

Plot[{
    NIntegrate[ PDF[ MultinormalDistribution[
      {0, 0}, ({{1, 37/40}, {37/40, 1}})], {x, y}],
      {x, -∞, 0}, {y, -∞, u}],
    CDF[dist, {0, u}]},
  {u, 0, 1}]

Plot[
  CDF[ MultinormalDistribution[
      {0, 0}, ({{1, SetPrecision[x, 15]}, {SetPrecision[x, 15], 1}})],
    {0, 0.2`15}] -
  CDF[MultinormalDistribution[{0, 0}, ({{1, x}, {x, 1}})],
    {0, 0.2}],
  {x, 0.8, 1}, PlotRange -> All]

dist = MultinormalDistribution[{0, 0}, ({{1, 37/40}, {37/40, 1}})];

cdf1 = CDF[dist, {0, .2}]

(*  0.446357  *)

cdf2 = CDF[dist, {0, .2`15}]

(*  0.47007303261424  *)

ContourPlot[
 CDF[dist, {x, y}], {x, -3, 3}, {y, -3, 3},
 PlotRange -> All,
 MaxRecursion -> 3,
 WorkingPrecision -> 15]