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. $$f(x)=2 x^{2} \text { on }[1,6] ; n=5$$

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.
$$f(x)=2 x^{2} \text { on }[1,6] ; n=5$$

Key Concepts

Underestimation and Overestimation of Areas

When using Riemann sums, the choice between left and right endpoints can lead to systematic underestimation or overestimation of the true area under a curve. For instance, if a function is increasing, a left sum tends to underestimate the area while a right sum tends to overestimate it. Understanding these tendencies helps in assessing the accuracy of the approximation and in choosing methods for better estimates.

Right Riemann Sum

The right Riemann sum approximates the area under a curve by using the right endpoint of each subinterval for function evaluation. Similarly, it constructs rectangles with heights given by the function's value at the end of each subinterval. Like the left Riemann sum, this can also result in an under or overestimate relative to the exact value, contingent upon whether the function is increasing or decreasing.

Riemann Sums

Riemann sums are a method to approximate the definite integral of a function by dividing the area under its curve into simple shapes (usually rectangles) and summing their areas. This technique forms the foundation of integral calculus and provides a practical approach to understanding how integration approximates area.

Partitioning the Interval and ?x

Partitioning the interval involves dividing the given interval into smaller subintervals of equal width. The width of each subinterval, denoted as ?x, is calculated by the formula ?x = (b - a)/n, where a and b are the endpoints of the interval and n is the number of subintervals. This concept is essential for determining the sample points at which the function is evaluated.

Left Riemann Sum

The left Riemann sum approximates the area under a curve by using the left endpoint of each subinterval to evaluate the function values. It essentially constructs rectangles whose heights are determined by the value of the function at the beginning of each subinterval. This method can provide an underestimate or overestimate of the true area depending on the nature of the function.