+- HP Forums (https://www.hpmuseum.org/forum)
+-- Forum: Not HP Calculators (/forum-7.html)
+--- Forum: Not remotely HP Calculators (/forum-9.html)
+--- Thread: Problem with latest XKCD (/thread-22115.html)
Problem with latest XKCD - KeithB - 08-02-2024 05:39 PM
One of the numbers asked for in the latest XKCD is the integral from 0 to pi of x*sin^2 x.
https://xkcd.com/2966/
Explain xkcd says the answer is pi^2/4 or 2.4674...
https://www.explainxkcd.com/wiki/index.php/Main_Page
I tried with the HP71: Integral(0, pi, 1e-12, Ivar*sin(Ivar)*sin(Ivar)) = .007413..
The prime in cas mode gives an error
RE: Problem with latest XKCD - KeithB - 08-02-2024 06:05 PM
never mind. I had degrees mode set.
RE: Problem with latest XKCD - Dave Britten - 08-02-2024 06:35 PM
Okay, that "Game Theory" one definitely got a chuckle out of me.
RE: Problem with latest XKCD - Albert Chan - 08-02-2024 08:25 PM
We can solve definite integral by folding:
\(\int_a^b f(x) \;dx =
\int_a^\frac{a+b}{2} (f(x) + f(a+b-x)) \;dx
\)
∫(x*sin(x)^2, x=0 .. pi) = ∫(x*sin(x)^2 + (pi-x)*sin(x)^2, x=0 .. pi/2) = pi * ∫(sin(x)^2, x = 0 .. pi/2)
We can use integral identities for RHS ... or we just fold again!
pi * ∫(sin(x)^2, x = 0 .. pi/2) = pi * ∫(sin(x)^2 + cos(x)^2, x = 0 .. pi/4) = pi * pi/4 = pi^2/4