To find the area of the shaded region, we need to:

1. Identify the functions that form the boundaries of the region.

   • The curves are:
     • $y = x^2$ (a parabola)
     • $y = 8 - 2x$ (a straight line)
     • $y = \frac{2x}{3} + \frac{16}{3}$ (another straight line)

2. Find the points of intersection between the curves, which are already marked as $(-2, 4)$, $(1, 6)$, and $(2, 4)$.

3. Set up the integrals to compute the area.

   • The area can be computed by splitting it into two regions:
     1. From $x = -2$ to $x = 1$, the area between the line $y = 8 - 2x$ (top) and the parabola $y = x^2$ (bottom).
     2. From $x = 1$ to $x = 2$, the area between the line $y = \frac{2x}{3} + \frac{16}{3}$ (top) and the parabola $y = x^2$ (bottom).

Step 1: Calculate the area for $x \in [-2, 1]$

$A_1 = \int_{-2}^{1} \left( (8 - 2x) - x^2 \right) dx$

Step 2: Calculate the area for $x \in [1, 2]$

$A_2 = \int_{1}^{2} \left( \left( \frac{2x}{3} + \frac{16}{3} \right) - x^2 \right) dx$

Step 3: Total Area

$A = A_1 + A_2$

I will now compute these integrals to find the total area.The total area of the shaded region is $28$ square units.

Would you like further details on any specific step or have any questions?

Here are 5 related questions for deeper understanding:

1. How do you determine points of intersection between curves?
2. What are the general steps for setting up an area between two curves?
3. Why is it necessary to split the region into two integrals in this case?
4. How would you approach this problem if the curves were more complex?
5. Can you use numerical integration methods when functions are not easily integrable?

Tip: Always sketch the region and label the intersection points clearly before setting up integrals for finding areas. This ensures you select the correct boundaries for your integration limits.