📚 10B Definite integration and the area under a curve | 定积分与曲线下面积
In calculus, differentiation measures the instantaneous rate of change. Its reverse process, integration, began as a way to find areas under curves. The definite integral gives a precise numerical value for that area – or, more generally, the net accumulation of a quantity over an interval. This concept turns the geometry of curves into a powerful algebraic tool, linking antiderivatives to actual measurements. Mastering it is essential for everything from probability to physics, and it sits at the heart of IB mathematics.
在微积分中，求导衡量的是瞬时变化率。而它的逆过程——积分，最初是为了计算曲线下的面积而发展的。定积分能够为这一面积提供一个精确的数值，更广义地说，是度量某个量在一个区间上的净累积。这一概念将曲线的几何性质转化为强大的代数工具，并把反导函数和实际测量结果紧密联系起来。掌握它对于从概率到物理学的诸多领域都至关重要，也是 IB 数学的核心内容之一。
1. From rate of change to accumulation | 从变化率到累积量
ve the total accumulated change. Geometrically, when f(x) ≥ 0, the definite integral ∫ab f(x) dx exactly equals the area of the region bounded by the graph of y = f(x), the x‑axis, and the vertical lines x = a and x = b.
导数告诉我们一个量在某一瞬间变化得有多快。反过来想，不定积分从变化率还原出原来的函数。而定积分则延伸了这一思路：它将微小的变化量 f(x)·dx 在区间 [a, b] 上累加起来，得到总的累积变化。从几何上看，当 f(x) ≥ 0 时，定积分 ∫ab f(x) dx 恰好等于曲线 y = f(x)、x 轴以及竖直线 x = a 和 x = b 所围成区域的面积。
This connection between accumulation and area is the reason the definite integral appears everywhere: the distance travelled by an object (area under a velocity graph), the total growth of a population (area under a growth‑rate curve), or the work done by a varying force. In IB exams, you will be asked to interpret a definite integral both as a number and as a geometric or physical quantity.
累积与面积之间的这种联系，正是定积分无处不在的原因：物体运动的路程（速度图像下方的面积）、种群的总增长量（增长率曲线下的面积），或者变力所做的功。在 IB 考试中，你既需要将定积分理解为一个数值，也需要将其解释为几何量或物理量。
2. Riemann sums and the definite integral | 黎曼和与定积分
artition [a, b] into n subintervals, each of width Δx = (b − a)/n. Choose a sample point xi* in the i‑th subinterval. The Riemann sum Σi=1n f(xi nd the width Δx → 0, the limit of these sums, if it exists, is the definite integral ∫ab f(x) dx.
定积分的正式定义建立在将区域切分成 n 个薄矩形这一想法之上。将区间 [a, b] 分割成 n 个子区间，每个子区间宽度为 Δx = (b − a)/n。在每个子区间内选取一个样本点 xi*。黎曼和 Σi=1n f(xi*) Δx 便近似曲线下的有号面积。当矩形个数趋于无穷（n → ∞）且宽度 Δx → 0 时，这些和的极限若存在，就是定积分 ∫ab f(x) dx。
| Type of Riemann sum (English) | 黎曼和类型（中文） |
|---|---|
| Left‑endpoint sum: xi* = xi−1 | 左端点求和：xi* = xi−1 |
| Right‑endpoint sum: xi* = xi | 右端点求和：xi* = xi |
| Midpoint sum: xi* = (xi−1 + xi)/2 | 中点求和：xi* = (xi−1 + xi)/2 |
For continuous functions, the limit exists and is independent of the choice of sample points. This is the reason we can later use antiderivatives to evaluate the integral exactly, rather than computing endless sums.
对于连续函数，该极限存在且与样本点的选取无关。这便是我们之后能够使用反导函数来精确计算积分，而不必进行无穷求和的原因。
3. Notation and terminology | 记法与术语
The notation ∫ab f(x) dx is read “the integral from a to b of f(x) with respect to x”. The elongated S, ∫, was introduced by Leibniz to signify a continuous sum. The lower limit a and the upper limit b set the boundaries of integration. The function f(x) is the integrand, and the symbol dx reminds us that the variable of integration is x and that the ‘strips’ have width dx.
∫ab f(x) dx 这一记法读作“从 a 到 b 对 f(x) 关于 x 的积分”。拉长的 S 形符号 ∫ 由莱布尼茨引入，表示连续求和。下限 a 与上限 b 划定了积分区间；函数 f(x) 称为被积函数。符号 dx 则提示我们积分变量是 x，且每个“条带”的宽度为 dx。
When the interval is reversed, the sign changes: ∫ab f(x) dx = − ∫ba f(x) dx. If the upper and lower limits are equal, the integral is zero. These properties follow directly from the definition and are often useful when tackling definite integrals with variable limits.
当区间反向时，符号会改变：∫ab f(x) dx = − ∫ba f(x) dx。若上、下限相等，则积分为零。这些性质直接源于定义，在处理带有可变上下限的定积分时常常非常有用。
4. The Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem of Calculus (FTC) is the bridge between differential and integral calculus. It has two parts, both essential for IB.
微积分基本定理（FTC）是连接微分学与积分学的桥梁。它包含两个部分，对 IB 课程都至关重要。
| Part (English) | 部分（中文） | Formula / Meaning |
|---|---|---|
| Part 1 (Evaluation Theorem) | 第一部分（求值定理） | If F'(x) = f(x), then ∫ab f(x) dx = F(b) − F(a). (中文：若 F'(x) = f(x)，则 ∫ab f(x) dx = F(b) − F(a)。) |
| Part 2 (Differentiation of an integral) | 第二部分（积分上限函数求导） | If G(x) = ∫ax f(t) dt, then G'(x) = f(x). (中文：若 G(x) = ∫ax f(t) dt，则 G'(x) = f(x)。) |
Part 1 tells us that to find the definite integral, we simply evaluate any antiderivative at the limits and subtract. This is why the constant of integration (+ C) is not needed in definite integrals. Part 2 guarantees that differentiation and integration are inverse operations, and it is key when a definite integral depends on a variable upper limit.
第一部分告诉我们，要求定积分，只需在上下限处计算任一反导函数的值，然后相减即可。这就是为什么定积分中不需要积分常数 (+ C)。第二部分则保证微分与积分互为逆运算，当定积分的上限为变量时，这一部分尤为关键。
5. Evaluating definite integrals | 计算定积分
Once an antiderivative F(x) is known, the evaluation becomes straightforward: substitute the upper limit, substitute the lower limit, and compute F(b) − F(a). Always write the antiderivative in square brackets with the limits before substituting,
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