"Things You Wouldn't Know If We Didn't Blog Intermittently."
14 March 2022
The free wi-fi password is...
I've written a possible answer in the first comment on this post, but several readers have come up with a better answer than mine (assuming that there is a typo in the equation as written)... (via)
The amazing coincidence is that if you ignore the dx under the square root as a typo, the integral is equal to exactly pi, and you posted it on March 14.
(If you split the integral into two distributing the sum, the first looks really complicated but it's zero since it's an odd function integrated over a symmetrical interval; the second is just half the area of a semicircle of radius 2, so pi.)
Do you really think that's what they meant? Considering the publication date, you might suspect that the value is a familiar constant. And in fact it is. Since x^3 cos(x/2) is an odd function and the interval is symmetric about 0, you can just ignore all of that. What remains can be interpreted as 1/4 the area of a circle of radius 2. And, yes, ugh, putting the radical sign over the dx is an ugly typo.
The only catch about the password being Pi is: to how many decimal places does it have to be to be accepted as a valid password? Since Pi is an irrational number, the password could be really, reaaaalllllyyyy long.
It's "thefirstdigitsoftheanswer".
ReplyDeleteMaybe. See comments below.
DeleteThe dx shouldn't be under the roof of the √
ReplyDeleteMy eye was twitching slightly
DeleteThe amazing coincidence is that if you ignore the dx under the square root as a typo, the integral is equal to exactly pi, and you posted it on March 14.
ReplyDelete(If you split the integral into two distributing the sum, the first looks really complicated but it's zero since it's an odd function integrated over a symmetrical interval; the second is just half the area of a semicircle of radius 2, so pi.)
Do you really think that's what they meant? Considering the publication date, you might suspect that the value is a familiar constant. And in fact it is. Since x^3 cos(x/2) is an odd function and the interval is symmetric about 0, you can just ignore all of that. What remains can be interpreted as 1/4 the area of a circle of radius 2. And, yes, ugh, putting the radical sign over the dx is an ugly typo.
ReplyDelete"Speak, friend, and enter."
ReplyDeleteThe only catch about the password being Pi is: to how many decimal places does it have to be to be accepted as a valid password? Since Pi is an irrational number, the password could be really, reaaaalllllyyyy long.
ReplyDelete