The approximate area under the curve and above the x-axis on the interval [2, 5] using 6 rectangles with midpoints is 38.15 square units.
To approximate the area under the curve and above the x-axis on the interval [2, 5] using 6 rectangles, you can use either left rectangles, right rectangles, or midpoints. The appropriate choice depends on the shape of the curve and the desired accuracy of the approximation.
If the curve is increasing on the interval [2, 5], then using left rectangles will underestimate the true area. Using right rectangles will overestimate the true area. Using midpoints will give a more accurate approximation.
If the curve is decreasing on the interval [2, 5], then using left rectangles will overestimate the true area. Using right rectangles will underestimate the true area. Using midpoints will give a more accurate approximation.
In this case, the curve is increasing on the interval [2, 5].
Therefore, using midpoints will give a more accurate approximation of the area under the curve and above the x-axis.
The width of each rectangle is (5 - 2) / 6 = 0.5.
The midpoints of the rectangles are 2.25, 2.75, 3.25, 3.75, 4.25, and 4.75.
The areas of the rectangles are (2.25)^2 * 0.5 = 2.53125, (2.75)^2 * 0.5 = 3.828125, (3.25)^2 * 0.5 = 5.234375, (3.75)^2 * 0.5 = 6.7421875, (4.25)^2 * 0.5 = 8.453125, and (4.75)^2 * 0.5 = 11.3671875. The total area under the curve and above the x-axis is approximately 2.53125 + 3.828125 + 5.234375 + 6.7421875 + 8.453125 + 11.3671875 = 38.15625.
Therefore, the approximate area under the curve and above the x-axis on the interval [2, 5] using 6 rectangles with midpoints is 38.15 square units.