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. $$ L_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2] $$

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.
$$
L_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2]
$$

Key Concepts

Riemann Sum

A Riemann sum is a method used in calculus to approximate the area under a curve by summing the areas of multiple rectangles. This concept forms the basis of definite integrals and provides a way to estimate the accumulation of quantities when an antiderivative is not readily available.

Right-Endpoint Approximation

The right-endpoint approximation is a specific type of Riemann sum where the value of the function is taken at the right end of each subinterval. This technique is used to approximate the area under a curve by multiplying the function value at the right endpoint of each interval by the width of the interval and then summing these products.

Partitioning the Interval

Partitioning the interval involves dividing the overall interval into a finite number of subintervals of equal or varying lengths. This step is essential in forming Riemann sums, as it determines the width of each rectangle used in the approximation process.

Subinterval Width

The subinterval width, commonly denoted as ?x, is calculated by dividing the length of the overall interval by the number of subintervals. It represents the horizontal length of each rectangle in the Riemann sum approximation, affecting the accuracy of the result.

Function Evaluation

Function evaluation in this context refers to the process of computing the value of the given function at specific sample points (in the case of a right-endpoint approximation, at the right endpoints of the subintervals). These values are critical as they serve as the heights of the rectangles used to estimate the area under the curve.

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