Let's divide the shaded region into two areas:
area 1: x = 0 ---> x = 2
ares 2: x = 2 ---> x = 4
In area 1, we need to find the area under g(x) = x and in area 2, we need to find the area between g(x) = x and f(x) = (x - 2)^2. Now let's set up the integrals needed to find the areas.
Area 1:
[tex]A\frac{}{1} = ∫g(x)dx = ∫xdx = \frac{1}{2} {x}^{2} | \frac{2}{0} = 2[/tex]
Area 2:
[tex]A\frac{}{2} = ∫(g(x) - f(x))dx[/tex]
[tex]= ∫(x - {(x - 2)}^{2} )dx[/tex]
[tex] = ∫( - {x}^{2} + 5x - 4)dx[/tex]
[tex]= ( - \frac{1}{3}{x}^{3} + \frac{5}{2} {x}^{2} - 4x) | \frac{4}{2}[/tex]
[tex] = 2.67 - ( - 0.67) = 3.34[/tex]
Therefore, the area of the shaded portion of the graph is
A = A1 + A2 = 5.34
