To estimate the area under the curve on the interval $[-5, -1]$ using the right-endpoint approximation with 2 rectangles, we need to divide the interval into two equal subintervals and use the function values at the right endpoints of these intervals.

Step-by-Step Solution:

1. Divide the Interval:

   • The interval $[-5, -1]$ has a width of $4$ units.
   • Dividing it into 2 subintervals gives a width of $\Delta x = 2$ for each rectangle.

2. Identify the Right Endpoints:

   • The subintervals are $[-5, -3]$ and $[-3, -1]$.
   • The right endpoints for these intervals are $x = -3$ and $x = -1$.

3. Evaluate the Function at the Right Endpoints:

   • From the graph, estimate the function values at $x = -3$ and $x = -1$.
   • Approximate values (from observation):
     • $f(-3) \approx 2$
     • $f(-1) \approx 4$

4. Calculate the Area of Each Rectangle:

   • Area of the first rectangle (for $[-5, -3]$): $f(-3) \cdot \Delta x = 2 \cdot 2 = 4$.
   • Area of the second rectangle (for $[-3, -1]$): $f(-1) \cdot \Delta x = 4 \cdot 2 = 8$.

5. Add the Areas to Get the Total Approximate Area:

   • Total area $\approx 4 + 8 = 12$.

Conclusion:

The estimated area under the curve on the interval $[-5, -1]$ using the right-endpoint approximation with 2 rectangles is approximately 12 square units.

Would you like a deeper breakdown of this process or have any additional questions?

Here are some related questions to explore further:

1. How does the right-endpoint approximation compare with the left-endpoint or midpoint approximation?
2. What happens to the accuracy of the approximation if we increase the number of rectangles?
3. How would you estimate the area if the intervals were unevenly spaced?
4. What is the significance of calculating areas under a curve in real-world applications?
5. How would this approximation change if we used a trapezoidal method instead?

Tip: When approximating areas, the more rectangles you use, the closer the approximation is to the actual area under the curve, as smaller intervals capture more details of the curve's shape.