Applications Involving the Integral as a Sum | 积分作为求和的应用

📚 Applications Involving the Integral as a Sum | 积分作为求和的应用

The definite integral is often introduced as a way to add up infinitely many small pieces. In AS Mathematics, this idea of the integral as a sum is applied to areas, motion, rates of change and other accumulated quantities. Understanding the link between summation and integration helps you interpret what an integral actually measures.

定积分通常被理解为把无穷多个微小部分相加的一种方法。在 AS 数学中，把积分视为求和的思想被应用于面积、运动、变化率以及其他累积量。理解求和与积分之间的联系，有助于你解释积分究竟在度量什么。

1. From Riemann Sums to Definite Integrals | 从黎曼和到定积分

A definite integral is defined as the limit of a Riemann sum. We split the interval [a, b] into n subintervals of width Δx, choose sample points xᵢ, form the sum Σ f(xᵢ) Δx, and let n → ∞. If the limit exists, it is written as ∫ₐᵇ f(x) dx.

定积分被定义为黎曼和的极限。我们把区间 [a, b] 分成 n 个宽度为 Δx 的小区间，选取样本点 xᵢ，构造和式 Σ f(xᵢ) Δx，并令 n → ∞。如果极限存在，就写成 ∫ₐᵇ f(x) dx。

In this notation, f(x) dx represents one typical ‘infinitesimal piece’, and the integral sign is an elongated S for ‘sum’. The lower limit a and upper limit b tell us where the summation starts and ends.

在这个记号中，f(x) dx 代表一个典型的“无穷小片段”，积分号是拉长的 S，表示“求和”。下限 a 和上限 b 告诉我们求和从哪里开始、到哪里结束。

2. Area Under a Curve | 曲线下的面积

If f(x) ≥ 0 on [a, b], the sum of thin rectangle areas f(x) Δx approximates the area under y = f(x). Taking the limit gives the exact area: A = ∫ₐᵇ f(x) dx.

如果 f(x) ≥ 0 在 [a, b] 上成立，那么一系列细长方形面积 f(x) Δx 的和就近似于 y = f(x) 下方的面积。取极限后得到精确面积：A = ∫ₐᵇ f(x) dx。

When f(x) takes negative values, the definite integral gives a signed or net area: regions above the x-axis count positively, and regions below count negatively.

当 f(x) 取负值时，定积分给出的是带符号面积或净面积：x 轴上方的区域计为正，x 轴下方的区域计为负。

3. Area Between Two Curves | 两条曲线之间的面积

If f(x) ≥ g(x) on [a, b], the area between the curves is obtained by summing vertical strips of height f(x) − g(x) and width dx. Thus the area is A = ∫ₐᵇ [f(x) − g(x)] dx.

如果在 [a, b] 上 f(x) ≥ g(x)，两条曲线之间的面积可以通过将高度为 f(x) − g(x)、宽度为 dx 的竖直窄条相加得到。因此面积为 A = ∫ₐᵇ [f(x) − g(x)] dx。

The limits a and b are usually the x-coordinates of the intersection points. If the curves cross, split the interval and use |f(x) − g(x)| to count the actual area.

上下限 a 和 b 通常是交点横坐标。如果两条曲线相交，需要拆分区间并使用 |f(x) − g(x)| 来计算实际面积。

4. Displacement from Velocity | 由速度求位移

Velocity is the rate of change of displacement with respect to time: v(t) = ds/dt. Summing small displacements v(t) dt over a time interval gives the net change in position: displacement = ∫ v(t) dt.

速度是位移对时间的变化率：v(t) = ds/dt。把每个小时刻的微小位移 v(t) dt 在一段时间内相加，就得到位置的总变化量：位移 = ∫ v(t) dt。

Total distance travelled is different: it is found by integrating speed, not velocity. Distance = ∫ |v(t)| dt, because distance cannot be negative.

总路程则不同：它由速率而不是速度积分得到。路程 = ∫ |v(t)| dt，因为路程不能为负。

5. Velocity from Acceleration | 由加速度求速度

Similarly, acceleration is the derivative of velocity: a(t) = dv/dt. The change in velocity over [t₁, t₂] is the sum of all small changes a(t) dt, so Δv = v(t₂) − v(t₁) = ∫ v'(t) dt = ∫ a(t) dt.

类似地，加速度是速度的导数：a(t) = dv/dt。在 [t₁, t₂] 上速度的变化量就是所有微小变化 a(t) dt 的和，所以 Δv = v(t₂) − v(t₁) = ∫ v'(t) dt = ∫ a(t) dt。

To find the velocity function, add the initial velocity: v(t) = v(t₁) + ∫ a(t) dt. Without the initial condition, the integral only gives the change in velocity.

要求速度函数，需要加上初始速度：v(t) = v(t₁) + ∫ a(t) dt。没有初始条件时，积分只给出速度的变化量。

6. Total Change from a Rate of Change | 由变化率求总变化量

The Fundamental Theorem of Calculus says that ∫ₐᵇ F'(x) dx = F(b) − F(a). This is exactly the ‘sum of small changes’ idea: adding up all the instantaneous rates of change gives the net change in F.

微积分基本定理指出 ∫ₐᵇ F'(x) dx = F(b) − F(a)。这正是“微小

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