Let Ln denote the left-endpoint sum using n subintervals and...

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$ L_{8} \text { for } x^{2}-2 x+1 \text { on }[0,2] $$

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.
$$
L_{8} \text { for } x^{2}-2 x+1 \text { on }[0,2]
$$

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the definite integral of a function by summing the areas of simple shapes, usually rectangles. This method lays the groundwork for understanding integration, where the total area under a curve is estimated by adding up the areas of rectangles whose heights are determined by the function’s values at specific sample points.

Partitioning an Interval

Partitioning an interval involves dividing the full interval of integration into a number of smaller subintervals. This step is crucial for setting up a Riemann sum because it determines the base (or width) of each of the approximating rectangles.

Subinterval Width (Delta x)

Delta x represents the width of each subinterval and is calculated by dividing the total length of the interval by the number of subintervals. This uniform width is essential for accurately constructing the rectangles in a Riemann sum approximation.

Left-Endpoint Sum

The left-endpoint sum is a specific type of Riemann sum where the function’s value at the left endpoint of each subinterval is used to determine the height of each rectangle. This method results in an approximation of the integral and is especially useful for understanding how sample point selection affects the approximation.

Function Evaluation at Sample Points

In the context of Riemann sums, evaluating the function at selected sample points (such as the left endpoints in a left-endpoint sum) provides the necessary heights for the rectangles. This step is particularly straightforward for polynomial functions and is central to forming the sum that approximates the area under the curve.

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