The Definite Integral as a Limit of a Sum | 积分作为和式的极限与应用

📚 The Definite Integral as a Limit of a Sum | 积分作为和式的极限与应用

In IB Mathematics, the definite integral is not just an antiderivative; it is defined as the limit of a sum of many small pieces. This idea links discrete computing with continuous exactness and gives us powerful tools for geometry, physics and statistics.

在 IB 数学中，定积分不只是求导的逆运算，它本质上是一系列微小量的和的极限。这种“分割、近似、求和、取极限”的思想，将离散计算与连续精确值联系起来，也为面积、路程、平均值和体积等实际问题提供了统一方法。

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1. From Rectangle Sums to the Definite Integral | 从矩形和到定积分

Suppose a function f is continuous on the interval [a,b]. Divide [a,b] into n equal subintervals, each of width Δx = (b−a)/n. In the i-th subinterval choose a sample point xᵢ, and form the product f(xᵢ)Δx. Adding these products gives a Riemann sum that approximates the signed area under the curve.

设函数 f 在区间 [a,b] 上连续。将 [a,b] 分成 n 个等宽小区间，每个区间宽度为 Δx = (b−a)/n。在第 i 个小区间内取一点 xᵢ，构造乘积 f(xᵢ)Δx，再把这些乘积加起来，得到的就是曲线下方“有向面积”的黎曼和近似值。

∫ₐᵇ f(x) dx = limₙ→∞ ∑ᵢ₌₁ⁿ f(xᵢ)Δx, Δx = (b−a)/n

When n tends to infinity, the rectangles become thinner and thinner, and the sum approaches the definite integral. The integral sign itself resembles an elongated S, standing for “sum”.

当 n 趋于无穷大时，矩形越来越窄，和式的极限就是定积分。积分符号 ∫ 就像拉长的字母 S，正是 “Sum（和）” 的含义。

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2. Left, Right and Midpoint Rectangles | 左矩形、右矩形和中点矩形

For the same n, the choice of sample point changes the estimate. The three most common choices are left endpoints, right endpoints and midpoints.

对于同样的 n，取样点的位置会改变估计值。最常用的三种取法分别是左端点、右端点和中点。

• Left endpoints: Lₙ = Δx ∑ᵢ₌₁ⁿ f(a+(i−1)Δx)

  左端点：Lₙ = Δx ∑ᵢ₌₁ⁿ f(a+(i−1)Δx)

• Right endpoints: Rₙ = Δx ∑ᵢ₌₁ⁿ f(a+iΔx)

  右端点：Rₙ = Δx ∑ᵢ₌₁ⁿ f(a+iΔx)

• Midpoints: Mₙ = Δx ∑ᵢ₌₁ⁿ f(a+(i−1/2)Δx)

  中点：Mₙ = Δx ∑ᵢ₌₁ⁿ f(a+(i−1/2)Δx)

For f(x)=x², a=0, b=1 and n=4, the approximations are shown below.

对于 f(x)=x²，取 a=0，b=1，n=4，各近似值为：

Method	Sum	Value
Left	0.25(0² + 0.25² + 0.5² + 0.75²)	0.21875
Right	0.25(0.25² + 0.5² + 0.75² + 1²)	0.46875
Midpoint	0.25(0.125² + 0.375² + 0.625² + 0.875²)	0.328125
Exact	∫₀¹ x² dx	1/3 ≈ 0.3333

The midpoint method is often noticeably more accurate than left or right rectangles.

中点法通常比左矩形法或右矩形法更精确。

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3. The Formal Limit Definition | 定积分的正式极限定义

More generally, a Riemann sum is built from any partition of [a,b]. For each subinterval of length Δxᵢ, choose a sample point cᵢ. If the sum ∑f(cᵢ)Δxᵢ tends to a single finite number as the maximum subinterval width approaches zero, that number is the definite integral.

更一般地，黎曼和可以基于 [a,b] 的任意分割。对于每个长度为 Δxᵢ 的小区间，取样本点 cᵢ。当最大小区间宽度趋于零时，如果和式 ∑f(cᵢ)Δxᵢ 趋于唯一有限值，这个值就是定积分。

∫ₐᵇ f(x) dx = limₙ→∞ ∑ᵢ₌₁ⁿ f(a + i(b−a)/n) · (b−a)/n

Example: evaluate ∫₀¹ x² dx using the definition. Here f(x)=x², a=0, b=1, Δx=1/n and xᵢ = i/n.

例：用定义计算 ∫₀¹ x² dx。这里 f(x)=x²，a=0，b=1，Δx=1/n，xᵢ = i/n。

Sₙ = ∑ᵢ₌₁ⁿ (i/n)² · (1/n) = (1/n³) ∑ᵢ₌₁ⁿ i²

Using the formula ∑ᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6, we obtain Sₙ = n(n+1)(2n+1)/(6n³), whose limit is 1/3.

利用平方和公式 ∑ᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6，得到 Sₙ = n(n+1)(2n+1)/(6n³)，其极限为 1/3。

∫₀¹ x² dx = limₙ→∞ n(n+1)(2n+1)/(6n³) = 1/3

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4. Geometric Meaning: Signed Area | 几何意义：有向面积

If f(x) ≥ 0 on [a,b], then ∫ₐᵇ f(x) dx equals the area between the curve and the x-axis. If f(x) is sometimes negative, each rectangle contributes f(xᵢ)Δx, which may be negative, so the definite integral gives signed area.

若 f(x) ≥ 0，则 ∫ₐᵇ f(x) dx 等于曲线与 x 轴之间的面积。若 f(x) 在某些区间上为负，则每个矩形贡献 f(xᵢ)Δx 可能为负，因此定积分给出的是有向面积。

Total geometric area = ∫ₐᵇ |f(x)| dx

This distinction is essential in exam questions that ask for “area” rather than “integral”.

这一区别在考试中非常重要，因为题目若问 “面积”，而不是 “定积分”，就需要对 |f(x)| 积分。

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5. Application: Displacement and Total Distance | 应用：位移与总路程

Given a velocity function v(t), the net displacement from t=a to t=b is the integral of v(t). However, total distance travelled is the integral of |v(t)|, because distance must always accumulate positively.

已知速度函数 v(t)，从 t=a 到 t=b 的位移是 v(t) 的定积分；但实际总路程必须用 |v(t)| 的定积分，因为路程总是正向累积的。

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