Left and right Riemann sums Complete the following steps for...

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$ a. Sketch the graph of the function on the given interval. b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$ c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve. d. Calculate the left and right Riemann sums. $$f(x)=e^{x / 2} \text { on }[1,4] ; n=6$$

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=e^{x / 2} \text { on }[1,4] ; n=6$$

Key Concepts

Riemann Sums

Riemann sums are a fundamental concept in calculus used to approximate the area under a curve by summing the areas of simple geometric shapes, usually rectangles. This method serves as an approximation of the definite integral and lays the groundwork for understanding integration as the limit of sums over partitions of the interval.

Partitioning Intervals

Partitioning an interval involves dividing it into smaller, equally spaced subintervals. This process defines the width of each subinterval (commonly denoted as ?x) and establishes grid points along the interval, which are crucial for evaluating the function at specific points within each interval while approximating the area.

Left and Right Endpoint Methods

The left and right endpoint methods refer to the technique of choosing the function value at the left or right end of each subinterval to represent the approximate height of the rectangle. These strategies lead to different summations known as the left and right Riemann sums, each offering a distinct approximation of the area under the curve.

Overestimation and Underestimation

Overestimation and underestimation in the context of Riemann sums occur due to the nature of the function's behavior over each subinterval. Depending on whether the function is increasing or decreasing, one of the endpoint methods will capture a value that is consistently higher or lower than the actual function's values over that subinterval, leading to an approximation that is either too high (overestimate) or too low (underestimate).

Definite Integral Approximation

The use of Riemann sums to approximate a definite integral is a central idea in numerical integration. This approach relies on calculating the sum of areas of the approximating shapes defined over the partitioned intervals and provides a stepping-stone toward the concept of the exact area under a curve as the number of partitions increases indefinitely.

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