Suppose we want to find the area under the curve
𝑦
=
𝑓
(
𝑥
)
(pictured below) on the interval
[
0
,
4
]
using a Riemann sum with
4
rectangles.

Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Question

Suppose we want to find the area under the curve 
𝑦
=
𝑓
(
𝑥
)
 (pictured below) on the interval 
[
0
,
4
]
 using a Riemann sum with 
4
 rectangles.

Which of the following provides the most accurate approximation of the area under the curve on the given interval?

BlackTom AI · Accepted Answer

First, let's restate the problem to ensure it's clear: we want the area under the curve y = f(x) from x = 0 to x = 4 using a Riemann sum with 4 rectangles, and we are asked which method provides the most accurate approximation.
Since the answer choices aren’t listed here, we rely on the given correct option: midpoints. 
Consider how different Riemann-sum schemes approximate area with 4 subintervals of equal width. The interval [0,4] is divided into 4 subintervals of width Δx = (4-0)/4 = 1. 
- Left-endpoint rule uses f(0), f(1), f(2), f(3) as the rectangle heights. This method can be quite biased if the function is not monotone, and its error decreases linearly with n.
- Right-endpoint rule uses f(1), f(2), f(3), f(4). This often mirrors the left rule’s bias but in the opposite direction, and similarly benefits only linearly from increasing n.
- Trapezoidal rule averages consecutive endpoint heights, producing heights that approximate the area under linear interpolants between points; its error also generally scales with 1/n, and for moderately smooth f it can be quite good but depends on concavity.
- Midpoint rule takes the function value at the center of each subinterval, i.e., f(0.5), f(1.5), f(2.5), f(3.5), as the rectangle heights. For a broad class of well-behaved functions, the midpoint rule has a smaller error term (it is typically of order 1/n^2 for sufficiently smooth f), making it more accurate than the left, right, and often the trapezoidal rule when the same number of subintervals is used.
A key idea behind the improved accuracy of the midpoint approach is that evaluating at the center of each subinterval tends to cancel out leading error terms that arise from the curvature of f, especially when f is continuous and has a bounded second derivative over [0,4].
If the function is highly nonlinear or exhibits rapid oscillations within a subinterval, any fixed-subinterval method’s accuracy can degrade, but among the common 4-rectangle options, the midpoint rule typically yields the best approximation for a broad range of smooth f.
In contrast, left- and right-endpoint schemes are more sensitive to the direction of the function’s slope over each subinterval, which can lead to consistent underestimation or overestimation depending on the function’s behavior. The trapezoidal rule, while often quite good, averages endpoints and can be less accurate than midpoint for functions with significant curvature within subintervals.
So, given the standard set of Riemann-sum options with 4 subintervals and a reasonably smooth f, the midpoint approach is commonly considered the most accurate among the basic methods, which aligns with the provided correct answer.

Suppose we want to find the area under the curve 𝑦 = 𝑓 ( 𝑥 ) (pictured below) on the interval [ 0 , 4 ] using a Riemann sum with 4 rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval? 单项选择题

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