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
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