The solution is later in the article.

> The insight that leads to a solution comes from the Jacobian.

It's not a unform half disk. It has more weight away from the Y axis.

You can imagine it's painted with watercolors and you want to collect the same ammount of ink.

In an uniform disk you have

    xx
   xxxx
   xxxx
  xxxxxx
  xxxxxx
  xxxxxx

    ..
   x..x
   x..x
   x..x
  Xx..xX
  Xx..xX
  Xx..xX