91Ó°ÊÓ

When dealing with intervals in calculus, especially when using Riemann Sums, dividing the main interval into smaller parts called "subintervals" is crucial. To find the width of each subinterval, use the formula \( \Delta x = \frac{b-a}{n} \). Here, \(a\) and \(b\) are the start and end of your interval, respectively, and \(n\) is the number of subintervals.
So, for the interval \([2, 6]\) and \(n = 4\), we have:

• The length of the entire interval is \((6-2) = 4\).
• Each subinterval, therefore, has a length of \( \Delta x = \frac{4}{4} = 1\).

Subintervals help break down complex calculations into simpler parts, paving the way for accurate estimation of areas under curves using Riemann Sums. This division is a fundamental step for integrating functions, allowing us to approximate the whole by examining its parts.

In the context of Riemann Sums, function evaluation is a vital step. It involves calculating the value of the function at various key points within each subinterval.
For this exercise, there are three main types of key points where function evaluation takes place:

• Left Endpoints: For the subintervals \([2, 3], [3, 4], [4, 5], [5, 6]\), the left endpoint evaluations are \(x_1^* = 2\), \(x_2^* = 3\), \(x_3^* = 4\), and \(x_4^* = 5\).
• Midpoints: For the same subintervals, the midpoints become \(x_1^* = 2.5\), \(x_2^* = 3.5\), \(x_3^* = 4.5\), and \(x_4^* = 5.5\).
• Right Endpoints: Here, you consider the right side of each subinterval for evaluation: \(x_1^* = 3\), \(x_2^* = 4\), \(x_3^* = 5\), and \(x_4^* = 6\).

Evaluating the function \(f(x) = 3x + 1\) at these points helps calculate the height of rectangles under the curve, leading to an estimation of the total area.

Graphical representation is an excellent way to visualize the Riemann Sum calculations. It involves sketching the graph of the function, and highlighting the areas under the curve that are approximated by rectangles.
Consider the function \(f(x) = 3x + 1\) over the interval \([2, 6]\). When plotting the function, you use:

• The slope \(3x\) plus the y-intercept, which is \(1\) in \(f(x)\).

For this exercise, draw rectangles whose heights are determined at the specified points (left, midpoint, right endpoints):

• Left endpoint approximation: Draw rectangles starting at each left endpoint within the interval.
• Midpoint approximation: Rectangles are drawn from the mid-points within each subinterval.
• Right endpoint approximation: Place rectangles beginning at each right endpoint of the subintervals.

By shading these rectangles under the line \(f(x)\) on the graph, you see how the sum of their areas approximates the total area under the curve between the given limits.