[psyang]

Originally Posted by SebC

full solution, I think, piggybacking on what bizaro86 has already explained about the first half of the function:

Spoiler!

A circle is x^2+y^2=r^2

y^2=r^2-x^2

y=sqrt(r^2-x^2)

y=sqrt(4-x^2) is a semi-circle of radius 2

integral of y=sqrt(4-x^2) from -2 to 2 is half of the area of a circle of radius 2 (because the bottom half is negative)

area of a circle is pi*r^2, in this case pi*4... half of that is pi*2

but the function we're integrating is y=1/2*sqrt(4-x^2), which is half as tall, so the area under the curve is also halved

so the answer to the full integral included the odd and even parts is pi

wifi password is 314 or something similar, though you still have to brute force it a bit because the cafe didn't say how many digits are required

which probably would've been in everyone's top few guesses anyways

psyang: What drives your interest in all this? Are you a math teacher professionally? I'm just curious.

We have a winner. Joint winner, I'd say, with Bizaro86 getting the first half, and SebC finishing it off. Bizaro86 did guess the right answer originally, but as always, you have to show your work

As for my interest? I just really enjoy math problems, and especially enjoy interesting problems with elegant solutions. I graduated with a math degree, though I work now as a software developer.

I loved high school and university math contests, and in my second or third year took a course called the 100 problems course. On the first day of class, we were given a sheet with 100 problems similar in nature to the ones I've posted in this thread. You tried to answer as many as you could by the end of the semester, and as long as you put in effort, you got an A, regardless of how many you solved (I think I ended up with solutions to about 60% of the problems). It was the first course where it clicked for me that university was a place to learn and explore interests, not just a place to try to get good grades and a degree.

The list of problems was lost during one of my many post university moves, something I still regret to this today. I guess this thread is my attempt to recreate the energy that was in those classes which consisted of about a dozen math geeks who had lots of fun discussing and solving problems together.