Signed Area as the Sum of Strips | 带符号面积：条带之和

📚 Signed Area as the Sum of Strips | 带符号面积：条带之和

In AS Mathematics, the definite integral is introduced through the idea of signed area. Before learning formal integration rules, you need to understand how the area between a curve and the x-axis can be built up from many thin rectangular strips. This article explains that fundamental picture: the signed area is the limit of a sum of strips.

在 AS 数学中，定积分通过带符号面积的思想引入。在学习正式积分法则之前，你需要理解曲线与 x 轴之间的面积如何由许多细矩形条带逐步构建。本文解释这个基本图景：带符号面积就是条带之和的极限。

1. What is Signed Area? | 什么是带符号面积？

When we talk about the region between a curve y = f(x) and the x-axis, we must decide how to treat parts below the x-axis. The signed area counts regions above the x-axis as positive and regions below the x-axis as negative. This is different from ordinary geometric area, which is always non-negative.

当我们讨论曲线 y = f(x) 与 x 轴之间的区域时，必须决定如何处理 x 轴下方的部分。带符号面积将 x 轴上方的区域计为正，下方的区域计为负。这与普通几何面积不同，几何面积总是非负的。

For example, if f(x) = x on the interval [-1, 1], the triangle above the x-axis gives positive area, while the triangle below gives negative area. The signed area is 0 because the two cancel, even though the total geometric area is 1.

例如，若 f(x) = x 在区间 [-1, 1] 上，x 轴上方的三角形给出正面积，下方的三角形给出负面积。带符号面积为 0，因为两者相互抵消，而总几何面积为 1。

2. The Basic Idea of Strips | 条带的基本思想

To estimate the signed area from x = a to x = b, we slice the region into n vertical strips of equal width. Each strip is approximately a rectangle, so its area is roughly height × width = f(xᵢ) Δx, where xᵢ is a chosen point inside the strip.

为了估计从 x = a 到 x = b 的带符号面积，我们把区域切成 n 个等宽的竖直条带。每个条带近似为一个矩形，所以其面积约为高 × 宽 = f(xᵢ) Δx，其中 xᵢ 是条带内选取的一个点。

Adding the areas of all n strips gives an approximation to the total signed area. The approximation improves as the strips become thinner, because each rectangle matches the curve more closely.

将所有 n 个条带的面积相加，得到总带符号面积的近似值。条带越细，近似越好，因为每个矩形与曲线的贴合度更高。

3. Partitioning the Interval [a, b] | 分割区间 [a, b]

An equal partition of [a, b] uses strip width Δx = (b – a) / n. The partition points are x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, xₙ = b. In general, xᵢ = a + i Δx for i = 0, 1, 2, …, n.

[a, b] 的等距分割使用条带宽度 Δx = (b – a) / n。分点为 x₀ = a，x₁ = a + Δx，x₂ = a + 2Δx，…，xₙ = b。一般地，xᵢ = a + i Δx，其中 i = 0, 1, 2, …, n。

Δx = (b – a) / n, xᵢ = a + i Δx

The i-th strip covers the subinterval [xᵢ₋₁, xᵢ], so its width is always Δx. This regular partition makes it easier to take limits later.

第 i 个条带覆盖子区间 [xᵢ₋₁, xᵢ]，因此其宽度始终为 Δx。这种规则分割使后面取极限更加方便。

4. Choosing the Height of a Strip | 选择条带高度

For the i-th strip on [xᵢ₋₁, xᵢ], the height can be chosen at the left endpoint f(xᵢ₋₁), the right endpoint f(xᵢ), or the midpoint f((xᵢ₋₁ + xᵢ)/2). These choices give different approximations, but all are valid Riemann sums.

对于第 i 个条带 [xᵢ₋₁, xᵢ]，高度可以取左端点 f(xᵢ₋₁)、右端点 f(xᵢ) 或中点 f((xᵢ₋₁ + xᵢ)/2)。这些选择给出不同的近似，但都是有效的黎曼和。

If f is increasing on [a, b], the left-endpoint sum underestimates the true area, while the right-endpoint sum overestimates it. If f is decreasing, the reverse is true.

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