📚 Areas Under Curves | 曲线下面积
Understanding how to find areas under curves is a fundamental application of integration in A-Level Mathematics. The definite integral allows us to compute the exact area enclosed between a curve, the x‑axis (or y‑axis), and given boundaries. However, careful handling is required when the curve crosses the axis or when finding regions between two curves. This chapter covers the core techniques and common pitfalls for Edexcel students.
理解如何计算曲线下面积是 A-Level 数学中积分应用的基本内容。定积分使我们能够准确计算曲线、x 轴（或 y 轴）与给定边界所围成的面积。然而，当曲线穿过坐标轴或需要求两条曲线之间的区域时，必须谨慎处理。本章涵盖 Edexcel 学生需要掌握的核心技巧和常见误区。
1. Definite Integrals as Area | 定积分表示面积
The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve y = f(x), the x‑ tric area. The Fundamental Theorem of Calculus links integration and differentiation: if F' ∫ₐᵇ f(x) dx = F(b) − F(a). This result provides a powerful tool to evaluate areas exactly.
定积分 ∫ₐᵇ f(x) dx 给出的是曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间的带符号面积。当 f(x) 在整个区间 [a, b] 上非负时，该积分直接代表实际的几何面积。微积分基本定理将积分与微分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。这一结果为精确计算面积提供了强大的工具。
It is essential to remember that the definite integral calculates a net area. Regions above the x‑axis contribute positively while regions below contribute negatively. Therefore, when a curve crosses the x‑axis, one must split the integral into separate intervals to obtain the total area.
必须记住，定积分计算的是净面积。x 轴上方的区域贡献正值，x 轴下方的区域贡献负值。因此，当曲线穿过 x 轴时，需要将积分分段，以求得总面积。
2. Area Bounded by a Curve and the x-axis | 曲线和 x 轴围成的面积
To find the area bounded by the curve y = f(x), the x‑is, and the lines x = a and x = b where f(x) ≥ 0 throughout, compute A = ∫ₐᵇ f(x) dx. Step‑by‑ r y = x² from x = 1 to x = 3 is ∫₁³ x² dx = [x³/3]₁³ = (27/3) − (1/3) = 26/3 square units.
求曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 所围成的面积，当 f(x) 在整个区间上非负时，计算 A = ∫ₐᵇ f(x) dx。步骤为：对 f(x) 积分求得原函数，代入上下限并相减。例如，y = x² 从 x = 1 到 x = 3 的下方面积为 ∫₁³ x² dx = [x³/3]₁³ = (27/3) − (1/3) = 26/3 平方单位。
Always check that f(x) does not change sign inside the interval. If it does, proceed to split the interval at the roots before integrating. A quick sketch helps visualise the problem and avoid sign mistakes.
务必检查 f(x) 在区间内是否变号。如果变号，应在根处分割区间后再积分。快速画出草图有助于直观理解并避免符号错误。
3. What Happens When the Curve Lies Below the x-axis? | 曲线位于 x 轴下方怎么办？
When part of the curve lies below the x‑ l points where f(x) = 0 within [a, b]. Split the integral at these roots. For each subinterval, evaluate the integral and take its absolute value. The total area is the sum of these absolute values. For instance, consider the area between y = x(x − 2) and the x‑s from x = 0 to x = 3. The curve crosses at x = 0 and x = 2. On [0,2], f(x) ≤ 0, so area = |∫₀² (x² − 2x) dx|; on [2,3], f(x) ≥ 0, area = ∫₂³ (x² − 2x) dx. Compute both and add.
当曲线的一部分位于 x 轴下方时，该子区间上的积分结果为负数。实际的面积是该积分的绝对值。实际操作中，先找出 f(x) = 0 在 [a, b] 内的所有根，并在这些根处将积分分段。对每个子区间计算积分，然后取其绝对值。总面积等于这些绝对值之和。例如，求 y = x(x − 2) 与 x 轴在 x = 0 到 x = 3 之间的面积。曲线
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