The problem involves calculating the area between two curves for each of the four diagrams in the image. Here's a step-by-step approach to address each question:

1. Identify the bounds of integration for each pair of curves: We need to determine where each pair of curves intersects to find the limits of integration.
2. Set up the area element, $\Delta L_i$: For each partition $i$, $\Delta L_i$ represents the differential element of area. This is typically given by the difference between the top and bottom functions over the interval of integration.
3. Write the integral expression: Integrate the difference between the upper and lower function over the interval to find the area.

Let's go through each case:

1. First Diagram (Top Left)

• Functions: $f(x) = 2 - x^2$ and $g(x) = x$.
• Intersection points: Solve $2 - x^2 = x$ to determine the bounds.
• Area element: $\Delta L_i = (f(x) - g(x)) \, dx$.
• Integral expression: $\text{Area} = \int_a^b \left( (2 - x^2) - x \right) \, dx$.

2. Second Diagram (Top Right)

• Functions: $f(x) = x + 4$ and $g(x) = x^2 - 2$.
• Intersection points: Solve $x + 4 = x^2 - 2$ to find the limits.
• Area element: $\Delta L_i = (f(x) - g(x)) \, dx$.
• Integral expression: $\text{Area} = \int_a^b \left( (x + 4) - (x^2 - 2) \right) \, dx$.

3. Third Diagram (Bottom Left)

• Functions: $y = x - 1$ and $x = 3 - y^2$.
• Rewrite in terms of $y$, if needed, or solve for the intersections in terms of $x$ and $y$.
• Integral expression will depend on the orientation; typically, in terms of $dy$ if integrating horizontally.

4. Fourth Diagram (Bottom Right)

• Functions: $y = \sqrt{x}$ and $y = 6 - x$.
• Intersection points: Solve $\sqrt{x} = 6 - x$ to find the bounds.
• Area element: $\Delta L_i = (6 - x - \sqrt{x}) \, dx$.
• Integral expression: $\text{Area} = \int_a^b \left( (6 - x) - \sqrt{x} \right) \, dx$.

Would you like me to solve the integrals for each case or go through the steps for finding the bounds of integration in more detail?

Questions for Further Understanding:

1. How do you find the intersection points of two functions?
2. What are the steps to set up an area integral between curves?
3. How do you determine if you should integrate with respect to $x$ or $y$?
4. Why is the upper function subtracted by the lower function in the area calculation?
5. How can we verify the correctness of the integral setup for each region?

Tip: When solving for intersection points, setting the functions equal to each other and simplifying often provides the limits of integration quickly.