The Correct Answer and Explanation is:
Based on the visual information in the graph, the correct answer is the Midpoint Riemann Sum with n=6 subintervals (M6).
Explanation
The image displays a graph of a function and illustrates a method for approximating the area under its curve, a process known as numerical integration. The technique shown is a specific type of Riemann sum. To identify which type, we must analyze how the representative rectangle is constructed.
A Riemann sum approximates the definite integral of a function by dividing the area into a series of rectangular strips. The total area is then estimated by summing the areas of these rectangles. The key difference between various Riemann sums lies in how the height of each rectangle is determined.
In the provided graph, the function is plotted over an x-axis interval that appears to be from 0 to 12. The problem likely uses six subintervals for the approximation, as suggested by the common notation “n=6” in calculus problems of this nature. This means the width of each subinterval, or each rectangle, is calculated as Δx = (12 – 0) / 6 = 2. The subintervals would be [0, 2], [2, 4], [4, 6], [6, 8], [8, 10], and [10, 12].
The graph highlights a single shaded rectangle over the subinterval from x = 6 to x = 8. We must observe where the top of this rectangle intersects the function’s curve to determine its height. The height is set by the function’s value at the midpoint of the interval [6, 8]. The midpoint is calculated as (6 + 8) / 2 = 7. Visually, the top center of the rectangle touches the curve precisely at x = 7.
This method, where the rectangle’s height is determined by the function’s value at the midpoint of its subinterval, is the definition of the Midpoint Riemann Sum. A Left Riemann Sum would have used the function’s value at x = 6 for the height, while a Right Riemann Sum would have used the value at x = 8. Since the sample point is x = 7, the graph depicts the Midpoint Riemann Sum.
