To sketch the region of integration for the given double integral, we need to identify the bounds:

1. Outer Integral (dx): From $x = 2$ to $x = 5$.
2. Inner Integral (dy):
   • Lower bound: $y = \frac{1}{3}x - \frac{8}{3}$
   • Upper bound: $y = \frac{2}{3}x + \frac{5}{3}$

Steps to Sketch the Region:

1. Plot the Vertical Lines: Draw vertical lines at $x = 2$ and $x = 5$.

2. Find Points on the Lines:

   • For $x = 2$:
     • Lower bound: $y = \frac{1}{3}(2) - \frac{8}{3} = \frac{2}{3} - \frac{8}{3} = -2$
     • Upper bound: $y = \frac{2}{3}(2) + \frac{5}{3} = \frac{4}{3} + \frac{5}{3} = 3$
   • For $x = 5$:
     • Lower bound: $y = \frac{1}{3}(5) - \frac{8}{3} = \frac{5}{3} - \frac{8}{3} = -1$
     • Upper bound: $y = \frac{2}{3}(5) + \frac{5}{3} = \frac{10}{3} + \frac{5}{3} = 5$

3. Plot the Lines:

   • Line 1 (lower bound): $y = \frac{1}{3}x - \frac{8}{3}$ can be rewritten in slope-intercept form.
   • Line 2 (upper bound): $y = \frac{2}{3}x + \frac{5}{3}$.

4. Determine Intersection Points: Find the intersection of the two lines: $\frac{1}{3}x - \frac{8}{3} = \frac{2}{3}x + \frac{5}{3}$ Solving this will give us the points to help close the region.

Vertices of the Region:

• Start with the points found from evaluating the lines at $x = 2$ and $x = 5$ and determine where they intersect.

Shading the Region:

Click on the vertices in order to outline and shade the area bounded by the two curves and the vertical lines.

If you'd like, I can assist further with visualizations or calculations regarding this integral. Would you like to proceed with that?

Questions to Consider:

1. What are the specific intersection points of the two lines?
2. How does the shape of the region affect the integral's value?
3. Would you like to calculate the area of the region of integration?
4. How do you interpret the physical significance of the function being integrated?
5. Do you need help with any other integrals or mathematical concepts?

Tip:

Always check the limits of integration carefully, as they define the region you are integrating over and can significantly affect the result.