Also in V10,

Integrate[1, {x, y} \[Element] Polygon[points]]
NIntegrate[Cos[x] Sin[y], {x, y} \[Element] Polygon[points]]

Hi, Christian,

If you have V10, then you can define more or less arbitrary regions, see documentation for ParametricRegion[] and ImplicitRegion[] as well.

points = {
   {-6.17, -33.75}, {-9.35, -33.75}, {-9.35, -33.78},
   {-9.35, -39.375}, {-9.35, -45.435}, {-9.35, -45.45},
   {-9.22, -45.45}, {-9.22, -39.375}, {-6.17, -39.375},
   {-6.17, -33.78}, {-6.17, -33.75}
   };
R = BoundaryMeshRegion[points, Line[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1}]]
Integrate[1, {x, y} \[Element] R]
NIntegrate[Cos[x] Sin[y], {x, y} \[Element] R]

I.M.

In general I think you will have to break up the region into convex regions. I don't know if there is a routine that will triangulate it. Maybe if you gave a practical but more generic case?

However, in the case you present it is quite simple.

vertexPoints = {{-6.17, -33.75}, {-9.35, -33.75}, {-9.35, -33.78},  {-9.35, -39.375}, {-9.35, -45.435}, {-9.35, -45.45}, {-9.22, -45.45},  {-9.22, -39.375}, {-6.17, -39.375}, {-6.17, -33.78}, {-6.17, -33.75}};

which plots as:

Graphics[
 {Line[vertexPoints],
  AbsolutePointSize[5], Point /@ vertexPoints},
 AspectRatio -> 1,
 PlotRange -> {{-10, -5}, {-50, -30}},
 Frame -> True,
 ImageSize -> 300]

The Boole function for the electrode region is given by:

inElectrode[x_, 
  y_] := (-9.35 <= x <= -6.17) \[And] (-39.735 <= 
     y <= -33.78) \[Or] (-9.35 <= x <= -9.22) \[And] (-45.45 <= 
     y <= -39.735)

which plots as:

RegionPlot[inElectrode[x, y], {x, -10, -5}, {y, -50, -30},
 PlotPoints -> 30]