Thanks for the investigation. I dig around a little and find another simple, and I would say even more troubling, example:
https://groups.google.com/g/sage-support/c/UIbQuoKLdKY/m/nYfF8UtHAgAJ Essential, the integral of a positive continuous function on a closed bounded interval integral(sqrt(cos(x)-cos(x)^3),x,0,pi/2) returns a negative number, in this case -2/3. Somehow SAGE chooses the negative root. Integrate the absolute value of the integrand produces another error. The command still returns a negative number even one slims down the interval a bit. I will go ahead a file a bug report. --Pong On Monday, May 1, 2023 at 7:29:32 AM UTC-7 Emmanuel Briand wrote: > This is not an answer but I have tried to reproduce this bug with a > simpler example and I obtained this: > > u(x) = sqrt((1-tan(x)^2)) > f(x) = pi/2-arccos(u(x)); > r(x) = f(x)*cos(x) > integral(r(x), (x, 0, pi/4)) > > 1.11072073453959 > > But > > > numerical_integral(r(x), 0, pi/4) > > (0.785398166410623, 6.110348980000323e-07) > > (Again an error of a factor sqrt(2)). > > > Strangely if you change f(x) into simply arccos(x) instead of pi/2-arccos(x), > a test seems to be peformed. > A warning is issued and no answer is given (the output after the warning is > again > integrate(arccos(sqrt(-tan(x)^2 + 1))*cos(x), x, 0, 1/4*pi) > ) > > u(x) = sqrt((1-tan(x)^2)) > f(x) = arccos(u(x)); > r(x) = f(x)*cos(x) > integral(r(x), (x, 0, pi/4)) > > Warning, integration of abs or sign assumes constant sign by intervals > (correct if the argument is real): > Check [abs(t_nostep)] > Warning, integration of abs or sign assumes constant sign by intervals > (correct if the argument is real): > Check [abs(t_nostep^2-1)] > Warning, integration of abs or sign assumes constant sign by intervals > (correct if the argument is real): > Check [abs(t_nostep^2-1)] > Warning, choosing root of > [1,0,%%%{4,[2,4]%%%}+%%%{-6,[2,2]%%%}+%%%{2,[2,0]%%%}+%%%{-6,[0,4]%%%}+%%%{10,[0,2]%%%}+%%%{-4,[0,0]%%%},0,%%%{4,[4,8]%%%}+%%%{-12,[4,6]%%%}+%%%{13,[4,4]%%%}+%%%{-6,[4,2]%%%}+%%%{1,[4,0]%%%}+%%%{4,[2,8]%%%}+%%%{-10,[2,6]%%%}+%%%{8,[2,4]%%%}+%%%{-2,[2,2]%%%}+%%%{1,[0,8]%%%}+%%%{-2,[0,6]%%%}+%%%{1,[0,4]%%%}] > at parameters values [49,-6] > Warning, choosing root of > [1,0,%%%{4,[2,4]%%%}+%%%{-6,[2,2]%%%}+%%%{2,[2,0]%%%}+%%%{-6,[0,4]%%%}+%%%{10,[0,2]%%%}+%%%{-4,[0,0]%%%},0,%%%{4,[4,8]%%%}+%%%{-12,[4,6]%%%}+%%%{13,[4,4]%%%}+%%%{-6,[4,2]%%%}+%%%{1,[4,0]%%%}+%%%{4,[2,8]%%%}+%%%{-10,[2,6]%%%}+%%%{8,[2,4]%%%}+%%%{-2,[2,2]%%%}+%%%{1,[0,8]%%%}+%%%{-2,[0,6]%%%}+%%%{1,[0,4]%%%}] > at parameters values [0,81] > Discontinuities at zeroes of t_nostep^2-1 were not checked > Discontinuities at zeroes of t_nostep^2-1 were not checked > Warning, integration of abs or sign assumes constant sign by intervals > (correct if the argument is real): > Check [abs(t_nostep)] > Error while checking exact value with approximate value, returning both! > > integrate(arccos(sqrt(-tan(x)^2 + 1))*cos(x), x, 0, 1/4*pi) > > Emmanuel > > > El dom, 30 abr 2023 a las 9:35, Pong (<wypo...@gmail.com>) escribió: > >> The codes >> >> x,y = var('x,y'); >> f(x) = acos(sqrt((1-tan(x)^2)/2)); >> g(x) = integral(sin(y)^4,(y,f(x),pi-f(x))); >> h(x) = sin(x)^2*cos(x)*g(x); >> integral(h(x),(x,-pi/4,pi/4)), numerical_integral(h(x),-pi/4,pi/4) >> >> produce >> >> (1/16*sqrt(2)*pi, (0.1963495451106892, 9.705160370278192e-07)) >> >> SageMath version: 9.8 on Ubuntu 22.04 (SAGE was complied from source) >> >> We believe the numerical answer is correct (that should be >> pi/16=0.1963....) since we got that answer by computing the integral in >> another way by hand. >> >> We were surprised that 'integral' can give us an answer and even more >> surprised by the fact that it is off by a factor of sqrt(2) from the answer >> given by 'numerical_integral'. >> >> Any insight of what's happening here? >> >> --Pong >> >> -- >> 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 sage-devel+...@googlegroups.com. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sage-devel/f1b3157d-20b5-4e7f-aa72-8046f84d5183n%40googlegroups.com >> >> <https://groups.google.com/d/msgid/sage-devel/f1b3157d-20b5-4e7f-aa72-8046f84d5183n%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> > -- 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 sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/0dbc7e0d-4619-461a-aca1-ec16a2379e72n%40googlegroups.com.
