Use a Riemann sum to approximate the area under the graph of...

Use a Riemann sum to approximate the area under the graph of $f(x)$ in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. (FIGURE CAN'T COPY)
$3 \leq x \leq 7, n=4,$ left endpoints

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Key Concepts

Approximating Rectangles

Approximating rectangles are the individual geometric shapes used in the Riemann sum process to estimate the area under a curve. The height of each rectangle is determined by the value of the function at a selected sample point within the corresponding subinterval, while the base is the width of the subinterval.

Left Endpoint Evaluation

Left endpoint evaluation is the method of choosing the leftmost point of each subinterval as the sample point at which to evaluate the function. This choice influences the height of the approximating rectangles and can impact the accuracy of the area estimation, especially for functions that are increasing or decreasing.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve by summing the areas of several simple shapes, typically rectangles, determined by the function's values at specific points within subintervals. This approach forms the basis for defining the definite integral in calculus.

Partitioning the Interval

Partitioning the interval involves dividing the domain of the function into a number of equal subintervals. The width of each subinterval, often denoted as ?x, is calculated by (b - a)/n, where [a, b] is the interval and n is the number of subintervals. This division is essential for applying numerical approximation methods.

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