I'm having the fallowing problem: I'm obtaining different results of the intersection area between two regions when I use the function "Area" and the function "NIntegrate". The results are just slightly different, but I know that there should not be any reason for that, since the numerical computation involved is not that complex. I know that the value returned by the "Area" function is the correct one, since it coincides in 15 significant figures with the numerical value given by the exact solution of the problem (I found it elsewhere). Below you can see my Mathematica code as well as a figure for the problem. The intersection area is that on gray color. Also, I don't know why the NIntegrate function is giving me the showed error :(
Rval = 10; Lval = 13; roVal = 2;
regBlue = ImplicitRegion[(Rval - roVal)^2 <= x^2 + y^2 <= Rval^2 && x > 0,{x,y}];
regRed = ImplicitRegion[(Rval - roVal)^2 <= x^2 + (y + Lval)^2 <= Rval^2 && x > 0, {x, y}];
regIntersection = RegionIntersection[regBlue, regRed];
N[Area[regIntersection], 15]
4.02362281375979
And the function NIntegrate gives:
NIntegrate[(\[Rho])* Boole[(Rval - roVal < \[Rho] < Rval) \[And] ((Rval - roVal)^2 < Lval^2 + \[Rho]^2 + 2 Lval \[Rho] Cos[\[Phi]] < Rval^2)], {\[Rho], 0, Rval}, {\[Phi], 0,\[Pi]}, AccuracyGoal -> 10, WorkingPrecision -> 15, MinRecursion -> 5, MaxRecursion -> 20]
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one ...
4.02314650566741
Additional information: In the function "NIntegrate" I tried to use polar coordinates with origin at point O. In the figure, the polar axis is pointing upward. The positive sense of variation for the polar angle [Phi] is clockwise .The distance from O to O' is Lval, the radius of both circles is Rval, and the thickness of the showed annular region in both circles is roVal . The value Lval^2 + [Rho]^2 + 2 Lval [Rho] Cos[[Phi]] is simply the square of the distance to O'. This problem is simple but is actually the root of a more serious problem I just faced in my research work. I tried for two hours to find the reason for this, but had no success.
Thanks in advance