Use the given function to find the [tex]y[/tex]-coordinate of [tex]A[/tex].
[tex]x=2 \implies y = 10+8\cdot2+2^2-2^3 = 22[/tex]
Find the equation of the line through the origin and [tex]A[/tex]. This line has slope
[tex]\mathrm{slope} = \dfrac{22-0}{2-0} = 11[/tex]
and passes through (0, 0), so its equation is
[tex]y - 0 = 11 (x-0) \implies y = 11x[/tex]
Then the area of the shaded region is given by the definite integral
[tex]\displaystyle \int_0^2 (y - 11x) \, dx = \int_0^2 (10 + 7x + x^2 - x^3) \, dx \\\\ ~~~~~~~~ = \left(10x + \frac72 x^2 + \frac13 x^3 - \frac14 x^4\right)\bigg|_0^2 \\\\ ~~~~~~~~ = 10\cdot2+\frac72\cdot2^2+\frac13\cdot2^3-\frac14\cdot2^4 = \boxed{\frac{98}3}[/tex]
