To find the area of the shaded region, we need to set up the definite integrals for the curves that bound the region and then compute the difference between them.

Step-by-step breakdown:

1. Identify the curves:

   • The parabolic curve is $y = x^2$.
   • The two linear equations are:
     • $y = 8 - 2x$ (left boundary),
     • $y = \frac{2x}{3} + \frac{16}{3}$ (right boundary).

2. Intersection points: The points where the curves intersect are important because they determine the limits of integration. The intersection points are given as:

   • $(-2, 4)$,
   • $(2, 4)$,
   • $(1, 6)$.

   So the boundaries for the region are from $x = -2$ to $x = 2$.

3. Set up the integral: The area of the shaded region is the difference between the linear functions (upper curves) and the parabolic function $y = x^2$ (lower curve). We'll split the integral into two parts because the upper boundary changes.

   The integral for the area is given by: $A = \int_{-2}^{1} \left( 8 - 2x - x^2 \right) dx + \int_{1}^{2} \left( \frac{2x}{3} + \frac{16}{3} - x^2 \right) dx$

Let me compute this for you.The area of the shaded region is $28$ square units.

Would you like a detailed breakdown of the calculation, or do you have any other questions?

Here are five related questions you might explore:

1. How do you find the area between two curves in general?
2. What is the significance of the intersection points when calculating area?
3. How would the result change if the region extended beyond $x = 2$?
4. Could you find the area using a different method, such as geometric shapes?
5. How would you handle cases where the functions intersect multiple times?

Tip: Always verify that the limits of integration match the region of interest by sketching the curves or checking their intersections!