Finding the value of the integral on this t-shirt

[rugerts]

Hello all,
I was watching some random videos on YouTube and found one of Dr. Ben Goertzel (a leading expert in the field of AI) and he was wearing the t shirt you see below. It caught my attention. I tried working it out, but I had no luck as it got messy fast. I then tried throwing it into Wolfram as well as another calculator and found an answer of 0.38. Is this correct? I would have thought that the answer would've 1/2, 1, or something along these lines to make sense of the joke.
To anyone who can work this out or explain it... thank you!
Here's Dr. Goertzel wearing the shirt as well as an enlarged view of the shirt itself.

 

[Orodruin]

Hint: That looks like its begging for some trigonometric substitution ...

 

[mathwonk]

guesses anyone?

 

[Hiero]

Hahah I believe the shirt has a typo!

I did the integral (it’s really not difficult with the right substitution) and the answer is certainly not 1/2

HOWEVER, I noticed, if the upperbound was 2/√π instead of √(2/π), then the answer would be 1/2

Kind of embarrassing! But maybe I’m just a glass $_0\int^{\frac{2}{\sqrt{\pi}}}(...)dx$ empty kind of guy.

 

[NascentOxygen]

Looks like Hiero has recognized the mistake.

http://m.wolframalpha.com/input/?i=integrate+sqrt(1/pi-(x-sqrt(1/pi))^2)+from+x=0+to+2/sqrt(pi)

 

[NascentOxygen]

Though I notice, too, that we can keep the limits as specified on the shirt, but change the left-most minus sign into a plus, and we'll again arrive at the 0.5 figure. Well, very close to it.

 

[rugerts]

Hahah I believe the shirt has a typo!

I did the integral (it’s really not difficult with the right substitution) and the answer is certainly not 1/2

HOWEVER, I noticed, if the upperbound was 2/√π instead of √(2/π), then the answer would be 1/2

Kind of embarrassing! But maybe I’m just a glass $_0\int^{\frac{2}{\sqrt{\pi}}}(...)dx$ empty kind of guy.

Very nice! I think you got it. Thanks so much.

 

[magadam]

0.5273