Brain Teaser - Area enclosed by a parabola and a line

[CR Haeger]

I tried to work the following problem fully by hand and then with CAS help from the HP Prime. Eventually I got to "solutions" on the Prime but it took a lot of variables creation and substitutions to get there. I am sure most here will make quick work of this.

The problem:
* Find the area enclosed by a function f(x)=x^2 and a line g(x) that intersects and is normal to f(x) for all x>0. Solve for area at x=+1.0.

* Write the function h(x) in terms of x only which describes the area enclosed by f(x) and g(x).

* Find the minimum value of h(x)

   

Best, Carl