On Sat, Oct 21, 2023 at 03:49:08PM -0700, Mild Shock wrote:
> The slope multiplied by the distance would be,
> using real-valued root (the Wolfram Alpha assumption):
>
> integral_0^(2 π) 1/100 (2 + cos(t)) cos(t)^(1/3) dt = 0.0364298
> https://www.wolframalpha.com/input?i=integ_0%5E%282*pi%29+cos%28t%29%5E%281%2F3%29+%282%2Bcos%28t%29%29+%2F100+dt&assumption=%22%5E%22+-%3E+%22Real%22
>
> But I am not sure whether the interval - π/2 to π/2,
> as you suggest, would give me the slope.
I suggested splitting interval into two parts: one where cos is
positive and one where it is negative. You then need to add
both parts, taking only one would be wrong.
> The factor
> cos(t)^(1/3) is indeed positive in this interval, but
>
> the integrand itself is not symmetric, for example the
> range π/2 to π is not a mirror of the range 0 to π/2. Also
> the smaller interval doesn't improve integrability:
>
> (1) -> integrate(cos(t)^(1/3)*(2+cos(t))/100, t = -%pi/2..%pi/2, "noPole")
>
> (1) "failed"
Yes, I was illustrating general principle. Above the approach
breaks down because FriCAS can not compute indefinite integral.
However, in cases when FriCAS can compute indefinite integral
you may have trouble due to singularities and choice of
brnaches. Splitting is a fairly general method to limit
troubles caused by singularities.
> Maybe there is a function in FriCAS for some
> numerical integral algorithm? Have to check the FriCAS
> manual, maybe I find something.
There is 'romberg' in NumericalQuadrature package.
--
Waldek Hebisch
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