Let a be the x coordinate of point a and b be the x coordinate of point B.

Then points a and b have coordinates (a, a? )(b，b？ )

The area between line AB and parabola is

[area of trapezoid]-? [a~b] x？ dx = 4/3

That is: (a? +b？ )(b-a)/2 - b？ /3 + a？ /3 = 4/3

Which can be simplified as: a? b/2 - a？ /6 + b？ /6 - ab？ /2 = 4/3

Which one can be broken down into: -(a-b)? /6 = 4/3

Solving the equation, we get b-a=2.

The coordinates of point p are (a, a? )(b，b？ )/2 = ((a+b)/2，(a？ +b？ )/2)

Substitute b-a=2 into ((a+b)/2, (a? +b？ ) /2), we get (b- 1, b? -2b+2)

Which is P(b- 1)=b? -2b+2

Let m be the x coordinate of point p, then m=b- 1, and b=m+ 1

P(m)=(m+ 1)？ -2(m+ 1)+2

P(m)=m？ + 1

Or y=x? + 1

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