Signed Area as the Sum of Strips | 有向面积：矩形条之和

📚 Signed Area as the Sum of Strips | 有向面积：矩形条之和

In AS-level mathematics, the definite integral is introduced as a way to measure the area between a curve and the x-axis. However, areas below the axis are treated as negative, leading to the concept of signed area. Understanding how this signed area can be built up from thin rectangular strips is fundamental to grasping integration.

在AS数学中，定积分用于度量曲线与x轴之间的面积。然而，x轴下方的面积被视为负值，这便引出了有向面积的概念。理解如何通过细长的矩形条累加成有向面积，是掌握积分的关键。

1. What is Signed Area? | 什么是有向面积？

xis. If f(x) ≥ 0 on [a, b], the result is the ordinary positive area. If f(x) ≤ 0, the integral gives the negative of the geometric area. When the curve crosses the axis, the integral represents the net signed area: the sum of areas above the axis minus the sum of areas below the axis.

在日常生活中，面积总是正的。然而，当我们用积分求函数 y = f(x) 从 x = a 到 x = b 下方（与x轴围成）的面积时，计算结果可能是正值、负值或零。这是因为积分考虑了曲线位于x轴上方还是下方。若在 [a, b] 上 f(x) ≥ 0，积分结果就是通常的正面积；若 f(x) ≤ 0，积分给出的是几何面积的相反数。当曲线穿过x轴时，积分表示净有向面积：x轴上方各区域面积之和减去x轴下方各区域面积之和。

2. Approximating Area with Strips | 用矩形条近似面积

To arrive at the exact signed area, we start by slicing the region into narrow vertical strips. Each strip is approximated by a rectangle whose width is a small Δx and whose height is determined by the function value at a chosen point within the strip. Summing the areas of these rectangles gives an approximation. As we make the strips thinner and more numerous, the approximation improves and eventually leads to the definite integral.

为了得到精确的有向面积，我们先将区域切成许多狭窄的竖直条带。每个条带近似看作一个矩形：宽度为很小的 Δx，高度取条带内某一点处的函数值。将这些矩形的面积求和，就得到近似值。随着条带越来越细、数量越来越多，近似程度不断提高，最终趋近于定积分的值。

广告 / Sponsored

3. Partition and Strip Width | 分割与条宽

idth. The width of each strip is Δx = (b − a)/n. The endpoints of these subintervals are x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, x_n = b. The i-th strip runs from x_i-1 to x_i. Using equal widths simplifies the summation, but strips do not have to be of equal width in general Riemann sums.

我们从闭区间 [a, b] 出发，将其等分成 n 个子区间。每个条带的宽度为 Δx = (b − a)/n。这些子区间的端点为 x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, x_n = b。第 i 个条带覆盖从 x_i-1 到 x_i 的范围。采用等宽条带可以简化求和，但在一般的黎曼和中，各条带宽度不必相等。

4. Choosing the Height of Each Strip | 选择每个矩形条的高度

For each strip, we need a sample point x*_i in the i-th subinterval [x_i-1, x_i] to determine the rectangle's height f(x*_i). Common choices include the left endpoint x_i-1, the right endpoint x_i, or the midpoint (x_i-1 + x_i)/2. Each choice leads to a different Riemann sum with its own properties of over- or underestimation.

对于每个条带，我们在第 i 个子区间 [x_i-1, x_i] 内选取一个样本点 x*_i，以 f(x*_i) 作为矩形的高。常见选取方式有左端点 x_i-1、右端点 x_i 或中点 (x_i-1 + x_i)/2。不同的选取方式对应不同的黎曼和，并具有高估或低估的特性。

5. Left and Right Riemann Sums | 左黎曼和与右黎曼和

The left Riemann sum L_n uses the left endpoint of each subinterval: L_n = Σ_i=1ⁿ f(x_i-1) Δx. If the function is increasing, L_n underestimates the true signed area; if decreasing, it overestimates.

左黎曼和 L_n 取每个子区间的左端点作为高：L_n = Σ_i=1ⁿ f(x_i-1) Δx。如果函数递增，左黎曼和会低估真实有向面积；如果函数递减，则会高估。

The right Riemann sum R_n uses the right endpoint: R_n = Σ_i=1ⁿ f(x_i) Δx. For an increasing function, R_n overestimates; for a decreasing function, it underestimates. By comparing L_n and R_n, we can bound the true value.

右黎曼和 R_n 取各子区间右端点：R_n = Σ_i=1ⁿ

Published by TutorHao | AS Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

备考资讯每日更新，关注不错过

考纲变化 · 科目分析 · 学习方法

在微信里长按二维码识别关注
或截图后用「扫一扫 → 相册」选择本图

公众号：tutorhao