📚 Areas under the x-axis | x 轴下方的面积
When calculating areas using integration, the definite integral does not always give a positive number. If the curve lies below the x-axis, the integral is negative, and the actual area must be found by taking the negative of that integral or by using absolute values. This article explains how to handle areas below the x-axis in Edexcel A-Level Mathematics.
使用积分计算面积时，定积分并不总是给出正数。如果曲线位于 x 轴下方，积分为负，实际面积需要取该积分的相反数或使用绝对值。本文讲解如何在 Edexcel A-Level 数学中处理 x 轴下方的面积。
1. The Definite Integral as Signed Area | 定积分作为有符号面积
he net signed area between the curve y = f(x) and the x-axis from x = a to x = b. The word "signed" ≤ 0 on [a, b], the integral will be negative, even though the geometric area is positive.
定积分 ∫ₐᵇ f(x) dx 度量的是曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的净有符号面积。"有符号"一词至关重要：x 轴上方的区域贡献正值，x 轴下方的区域贡献负值。因此，如果在 [a, b] 上 f(x) ≤ 0，即使几何面积为正，积分也会为负。
Net signed area = ∫ₐᵇ f(x) dx = (area above) − (area below)
2. Why the Integral Is Negative Below the x-Axis | 积分在 x 轴下方为何为负
In a Riemann sum, the integral is approximated by adding products f(xᵢ) Δx. When f(xᵢ) is negative, every term in the sum is negative, so the limit is negative. Geometrically, the height of each rectangle is measured from the x-axis to the curve, and below the axis this height is negative. Thus the integral records the orientation as well as the magnitude.
在黎曼和中，积分通过将 f(xᵢ) Δx 的乘积相加来近似。当 f(xᵢ) 为负时，和中的每一项
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