Part C

We start by looking at a plot of the original function.

Input := 


h =4; 

Input := 


f[x_] =x^3-4*x^2-x+h;

Input := 


Plot[f[x],{x,-2,3}]

Output =


-Graphics-

Confirm the area

Input := 


Integrate[f[x],{x,-1,1}]

Output =


16

--

3

We compute the area with the function shifted up by k.

Input := 


area[k_] = Integrate[f[x] + k,{x,-1,1}]

Output =


16

-- + 2 k

3

We find that the following value of k is appropriate.

Input := 


sol = Solve[area[k]==4,k]

Output =


        2/3

{{k -> 3   }}

And we confirm this.

Input := 


Integrate[f[x] + k/.sol[[1]],{x,-1,1}]

Output =


16      2/3

-- + 2 3

3