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Answer the last question first. In a word, it's a simple mistake: Cases[integrands[[-1]], Derivative[_], Infinity, Heads -> True] // Sort // Last Derivative[10] As you see, the highest order of derivative in integrands isn't tmax which equals 3. Another small mistake that leads to some warnings but doesn't influence the result is that ...


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Your Area results in a special function output with an assumption. You can force the evaluation simply by appending //N Area[2 Sqrt[(1 - p^2)^2 - r^2], {p, 0, 1}, {r, 0, 1 - p^2}] (* Integrate[-(-1 + p^2) Sqrt[1 + 16 p^2] EllipticE[-(3/(1 + 16 p^2))], {p, 0, 1}, Assumptions -> 0 < 1 - p^2] *) Area[2 Sqrt[(1 - p^2)^2 - r^2], {p, 0, 1}, {r, 0, 1 - ...


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How about something like Options[NArea] = Options[NIntegrate]; NArea[R__, ops:OptionsPattern[]] := Internal`InheritedBlock[{Integrate}, Unprotect[Integrate]; Integrate[Shortest[e__], r___Rule] := NIntegrate[e, ops]; Area[R] ] NArea[2 Sqrt[(1 - p^2)^2 - r^2], {p, 0, 1}, {r, 0, 1 - p^2}] 2.38377


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Put everything into SI units. Also, Mma appears to want "Kelvins" rather than "kelvins". $Version "10.0 for Mac OS X x86 (64-bit) (December 4, 2014)" h = UnitConvert[Quantity["PlanckConstant"]]; c = UnitConvert[Quantity["SpeedOfLight"]]; kb = UnitConvert[Quantity["BoltzmannConstant"]]; Iplanck[lambda_, T_] := Module[ {lambdaSI = ...



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