91Ó°ÊÓ

To begin understanding the right-endpoint sum, it's essential to know that this method is a way to approximate the area under a curve on a graph. For any given function over a specified interval, the right endpoints of subintervals are used to determine the heights of rectangles that approximate this area.

In the right-endpoint sum, each subinterval on the x-axis is determined, and the rightmost point of each subinterval is used. This point gives the x-value at which the function's value—or height of the rectangle—is calculated.

• The procedure involves dividing the main interval into smaller, equal parts.
• The function is evaluated at the right endpoint of each subinterval.
• These function values provide the height of each rectangle used in the approximation.

This method can be applied to different types of functions and intervals, providing a useful and straightforward way to estimate integrals.

One of the primary considerations when discussing Riemann sums, including the right-endpoint sum, is the width of each interval. Also known as the subinterval width, this is typically denoted as \(\Delta x\). It dictates how we divide our main interval into smaller segments.

The formula for calculating the interval width \(\Delta x\) is straightforward:\[ \Delta x = \frac{b-a}{n} \]where \(b\) and \(a\) are the endpoints of the interval, and \(n\) is the number of subintervals.

• A smaller \(\Delta x\) can yield a more accurate approximation as it means more subintervals.
• Larger \(\Delta x\) can quicken calculations but may result in less accurate approximations.

In our problem involving the interval \([2, 5]\) and 6 subintervals, \(\Delta x\) was calculated as \(\frac{1}{2}\). This precise determination is crucial for the subsequent steps in calculating the right-endpoint sum.

In the context of Riemann sums, determining the function's value at specific points is a critical step. These points are known as the evaluation points, and for a right-endpoint sum, they are the right edges of each subinterval.

Using the function given, \( f(x) = \frac{1}{x(x-1)} \), we evaluate this function at the right endpoints previously calculated. These evaluations give the height of each rectangle we're summing:

• Substitute each right endpoint into the function \( f \).
• These calculated values represent the heights of the rectangles over the interval \([2, 5]\).

These function values, when multiplied by the interval width, determine the contribution of each subinterval to the overall area approximation. Accuracy in this step directly ties to accuracy in the final sum.

Subintervals are the foundational components of dividing an interval to perform a Riemann sum estimation. In practice, the interval is sliced into smaller sections of equal width, called subintervals.

With each subinterval, a certain x-value from within it, typically a specific point like a left, middle, or right endpoint, is chosen for evaluating an approximation of the function's area over the subinterval.

• The number of subintervals, \(n\), directs how the main interval \([a, b]\) is partitioned.
• More subintervals can produce a more accurate approximation by better capturing the curve's behavior.

In this exercise, with the interval \([2, 5]\) divided into 6 subintervals, each had a width of \(\frac{1}{2}\). This setup is crucial for calculating an accurate right-endpoint sum. By partitioning the interval appropriately and choosing the correct evaluation points, we enhance the approximation’s fidelity.