Part A

We know that the roots of the function f will be ±Sqrt[h]. Therefore we want the definite integral of f between the roots to equal 2.

Input := 


area[h_] =Integrate[h-x^2,{x,-Sqrt[h],Sqrt[h]}]

Output =


   3/2

4 h

------

  3

Input := 


Solve[area[h]==2,h]

Output =


        3 2/3

{{h -> (-)   }}

        2

thus the correct vertical shift would be the cube root of 1.5 squared