Brain Teaser - Area enclosed by a parabola and a line

[Bunuel66]

Lower limit of integration:

Let P be a point on the curve. Coordinates of P are (x,x²).
Let M be a point on the straight line orthogonal to the tangent at P on the curve.

The vector T tangent at P has for coordinates (1,2x).
Then the vector PM is orthogonal to T.
With M of coordinates (xM, yM), if M is one the line one should have: (xM-x)+2x(yM-x²)=0.
This is just the dot product expressing the orthogonality.

When M is on the curve, yM=xM² then: (xM-x)+2x(xM²-x²)=0
As long as xM <> x: 1+2x(xM+x)=0
Solving for xM: xM=-x-1/2x=-(2x²+1)/2x

Regards