This integral turns out to be anything but trivial. Neither Sympy, Giac nor Fricas can solve it, in its original form nor in polar coordinates. BTW, neither Mathematica nor Rubi running on Mathematica can solve it in neither form.
The analytic answer given (semi-)trivially by Maxima is probably false. I'm unable to check it. It probably does not help, but maybe it might console... ;-). Le mercredi 13 mars 2019 18:33:10 UTC+1, [email protected] a écrit : > > Greetings, > > I've found that SageMath (8.3 and 8.6) gives me incorrect results when > integrating the simple integrand sqrt(1-x^2-y^2). The code that follows > illustrates the issue: > > banner() > > (x,y) = var('x y') > eps = var('eps',latex_name='\\epsilon') > assume(abs(x)<0.1, abs(y) < 0.1, abs(eps) < 0.1, eps > 0) > z = sqrt(1-x^2-y^2) > print(z) > > print('Integrating directly') > > int1 = z.integrate(x,-eps,eps).integrate(y,-eps,eps) > print(N(int1.substitute(eps==1e-4))) # Should be about 4e-8 > print(taylor(int1,(eps,0),4)) > > print('Integrating via Taylor series') > > int2 = taylor(z,(x,0),(y,0),8).integrate(x,-eps,eps).integrate(y,-eps,eps) > print(N(int2.substitute(eps==1e-4))) > print(taylor(int2,(eps,0),4)) > > Which prints (on sagecell.sagemath.org, to ensure a new version is being > tested): > > ┌────────────────────────────────────────────────────────────────────┐ > │ SageMath version 8.6, Release Date: 2019-01-15 │ > │ Using Python 2.7.15. Type "help()" for help. │ > └────────────────────────────────────────────────────────────────────┘ > sqrt(-x^2 - y^2 + 1) > Integrating directly > 1.99999998666667e-8 > -4/3*eps^4 + 2*eps^2 > Integrating via Taylor series > 3.99999998666667e-8 > -4/3*eps^4 + 4*eps^2 > > > Note the factor of 2 difference in the lowest-order term. > > > This may be an issue in one of the libraries called by Sage, but I'm not > experienced enough with the ecosystem to know which one (if any) could be at > fault. > > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
