(a) Use six rectangles to find estimates of each type for...

(a) Use six rectangles to find estimates of each type for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 12 $. (i) $ L_{6} $ (sample points are left endpoints) (ii) $ R_{6} $ (sample points are right endpoints) (iii) $ M_{6} $ (sample points are midpoints) (b) Is $ L_{6} $ an underestimate or overestimate of the true area? (c) Is $ R_{6} $ an underestimate or overestimate of the true area? (d) Which of the numbers $ L_{6} $, $ R_{6} $, or $ M_{6} $ gives the best estimate? Explain.

(a) Use six rectangles to find estimates of each type for the area under the given graph of $ f $ from
$ x = 0 $ to $ x = 12 $.
(i) $ L_{6} $ (sample points are left endpoints)
(ii) $ R_{6} $ (sample points are right endpoints)
(iii) $ M_{6} $ (sample points are midpoints)

(b) Is $ L_{6} $ an underestimate or overestimate of the true area?

(c) Is $ R_{6} $ an underestimate or overestimate of the true area?

(d) Which of the numbers $ L_{6} $, $ R_{6} $, or $ M_{6} $ gives the best estimate? Explain.

Key Concepts

Overestimate and Underestimate in Numerical Integration

In numerical integration, whether an approximation yields an overestimate or an underestimate depends on the behavior of the function over the interval. For example, if a function is increasing, the left endpoint approximation will tend to underestimate the area while the right endpoint will overestimate it. Understanding these tendencies is crucial for analyzing the error in an approximation and for choosing a method that best suits the characteristics of the function.

Right Endpoint Approximation

The right endpoint approximation uses the value of the function at the right endpoint of each subinterval to determine the height of the corresponding rectangle. Like the left endpoint method, this approximation can also cause consistent overestimation or underestimation based on the function’s behavior, which directly impacts how close the approximation is to the true area under the curve.

Midpoint Approximation

The midpoint approximation improves upon the left and right endpoint methods by using the value of the function at the midpoint of each subinterval as the height of the rectangle. This method often provides a more balanced estimate because it tends to average out the errors that occur from sampling only one endpoint in methods where the function is either increasing or decreasing.

Riemann Sums

Riemann sums are a fundamental concept in integral calculus, used to approximate the area under a curve. They work by partitioning the interval of interest into smaller subintervals, then summing up areas of simple shapes (usually rectangles) whose heights are determined by the function’s value at specific chosen points within each subinterval. This approach provides an estimation of the definite integral, and as the number of subintervals increases, the approximation generally improves.

Left Endpoint Approximation

The left endpoint approximation is a type of Riemann sum where the height of each rectangle is determined by the value of the function at the left endpoint of the subinterval. This method is straightforward to compute but can lead to systematic overestimation or underestimation of the actual area, depending on whether the function is increasing or decreasing on the interval.

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