Total circles area

You are encouraged to solve this task according to the task description, using any language you may know.
 Example circles
 Example circles filtered
Given some partially overlapping circles on the plane, compute and show the total area covered by them, with four or six (or a little more) decimal digits of precision. The area covered by two or more disks needs to be counted only once.

One point of this Task is also to compare and discuss the relative merits of various solution strategies, their performance, precision and simplicity. This means keeping both slower and faster solutions for a language (like C) is welcome.

To allow a better comparison of the different implementations, solve the problem with this standard dataset, each line contains the x and y coordinates of the centers of the disks and their radii   (11 disks are fully contained inside other disks):

         xc              yc          radius
     1.6417233788   1.6121789534   0.0848270516
    -1.4944608174   1.2077959613   1.1039549836
     0.6110294452  -0.6907087527   0.9089162485
     0.3844862411   0.2923344616   0.2375743054
    -0.2495892950  -0.3832854473   1.0845181219
     1.7813504266   1.6178237031   0.8162655711
    -0.1985249206  -0.8343333301   0.0538864941
    -1.7011985145  -0.1263820964   0.4776976918
    -0.4319462812   1.4104420482   0.7886291537
     0.2178372997  -0.9499557344   0.0357871187
    -0.6294854565  -1.3078893852   0.7653357688
     1.7952608455   0.6281269104   0.2727652452
     1.4168575317   1.0683357171   1.1016025378
     1.4637371396   0.9463877418   1.1846214562
    -0.5263668798   1.7315156631   1.4428514068
    -1.2197352481   0.9144146579   1.0727263474
    -0.1389358881   0.1092805780   0.7350208828
     1.5293954595   0.0030278255   1.2472867347
    -0.5258728625   1.3782633069   1.3495508831
    -0.1403562064   0.2437382535   1.3804956588
     0.8055826339  -0.0482092025   0.3327165165
    -0.6311979224   0.7184578971   0.2491045282
     1.4685857879  -0.8347049536   1.3670667538
    -0.6855727502   1.6465021616   1.0593087096
     0.0152957411   0.0638919221   0.9771215985

The result is   21.56503660... .



Related task

   Circles of given radius through two points.


See also

 http://www.reddit.com/r/dailyprogrammer/comments/zff9o/9062012_challenge_96_difficult_water_droplets/
 http://stackoverflow.com/a/1667789/10562


circles: @[
    @[    1.6417233788     1.6121789534 0.0848270516]
    @[neg 1.4944608174     1.2077959613 1.1039549836]
    @[    0.6110294452 neg 0.6907087527 0.9089162485]
    @[    0.3844862411     0.2923344616 0.2375743054]
    @[neg 0.2495892950 neg 0.3832854473 1.0845181219]
    @[    1.7813504266     1.6178237031 0.8162655711]
    @[neg 0.1985249206 neg 0.8343333301 0.0538864941]
    @[neg 1.7011985145 neg 0.1263820964 0.4776976918]
    @[neg 0.4319462812     1.4104420482 0.7886291537]
    @[    0.2178372997 neg 0.9499557344 0.0357871187]
    @[neg 0.6294854565 neg 1.3078893852 0.7653357688]
    @[    1.7952608455     0.6281269104 0.2727652452]
    @[    1.4168575317     1.0683357171 1.1016025378]
    @[    1.4637371396     0.9463877418 1.1846214562]
    @[neg 0.5263668798     1.7315156631 1.4428514068]
    @[neg 1.2197352481     0.9144146579 1.0727263474]
    @[neg 0.1389358881     0.1092805780 0.7350208828]
    @[    1.5293954595     0.0030278255 1.2472867347]
    @[neg 0.5258728625     1.3782633069 1.3495508831]
    @[neg 0.1403562064     0.2437382535 1.3804956588]
    @[    0.8055826339 neg 0.0482092025 0.3327165165]
    @[neg 0.6311979224     0.7184578971 0.2491045282]
    @[    1.4685857879 neg 0.8347049536 1.3670667538]
    @[neg 0.6855727502     1.6465021616 1.0593087096]
    @[    0.0152957411     0.0638919221 0.9771215985]
]
 
xMin: min map circles 'c -> c\0 - c\2
xMax: max map circles 'c -> c\0 + c\2 
yMin: min map circles 'c -> c\1 - c\2
yMax: max map circles 'c -> c\1 + c\2
 
boxSide: 500
 
dx: (xMax - xMin) / boxSide
dy: (yMax - yMin) / boxSide
 
count: 0
 
loop 0..dec boxSide 'r [
    y: yMin + r * dy 
    loop 0..dec boxSide 'c [
        x: xMin + c * dx
        loop circles 'circle [
            if (add (x - first circle)*(x - first circle) (y - circle\1)*(y - circle\1)) =< (last circle)*(last circle) [
                count: count + 1
                break
            ]
        ]
    ]
]
 
print ["Approximated area: " count * dx * dy]