To estimate the area between the x-axis and the graph of the function over the interval $[-1,5]$, we can approximate it by summing up the areas of the rectangles shown in the figure.

Each rectangle has a width of 1 unit, and the height is determined by the value of the function at the given x-coordinate. The shaded areas represent the rectangles, and some of them are above the x-axis, while others are below. Rectangles below the x-axis contribute negative areas, while those above contribute positive areas.

Here’s a step-by-step outline of the approach:

1. Width of rectangles: The width of each rectangle is $\Delta x = 1$ unit.

2. Heights of rectangles: The height of each rectangle corresponds to the function’s value at each x-coordinate:

   • For $x = -1$, the height is approximately $-4$.
   • For $x = 0$, the height is approximately $0$.
   • For $x = 1$, the height is approximately $3$.
   • For $x = 2$, the height is approximately $4$.
   • For $x = 3$, the height is approximately $2$.
   • For $x = 4$, the height is approximately $1$.
   • For $x = 5$, the height is approximately $0$.

3. Calculate the area of each rectangle:

   • At $x = -1$, the area is $1 \times (-4) = -4$.
   • At $x = 0$, the area is $1 \times 0 = 0$.
   • At $x = 1$, the area is $1 \times 3 = 3$.
   • At $x = 2$, the area is $1 \times 4 = 4$.
   • At $x = 3$, the area is $1 \times 2 = 2$.
   • At $x = 4$, the area is $1 \times 1 = 1$.
   • At $x = 5$, the area is $1 \times 0 = 0$.

4. Summing the areas: $\text{Total Area} = -4 + 0 + 3 + 4 + 2 + 1 + 0 = 6 \, \text{square units}.$

Thus, the estimated area is approximately $6 \, \text{units}^2$.

Would you like more details or clarification on this? Here are 5 questions related to this problem:

1. How do negative areas affect the total area calculation?
2. What is the significance of using rectangles to approximate the area?
3. Can the approximation method be improved, and if so, how?
4. How does the shape of the function impact the accuracy of the rectangular approximation?
5. What is the relationship between this method and the definite integral?

Tip: When using rectangles to estimate the area under a curve, more rectangles (i.e., a smaller width) will lead to a more accurate approximation.