He used the quadratic formula to get an equation in y, since you're integrating along the y-axis.
x²-2x+y=0
x = (-b ±√[b<sup>2</sup> - 4ac])/2a where a= 1 , b = -2 , c= y
EDIT: The alternative method is to complete the square:
-y = x<sup>2</sup> - 2x
-y =...