H - Definite Integrals | 定积分

📚 H - Definite Integrals | 定积分

The definite integral is a cornerstone of calculus, representing the accumulation of quantities such as area, volume, and total change. In the IB Mathematics HL course, students explore definite integrals through Riemann sums, the Fundamental Theorem of Calculus, various integration techniques, and rich applications including area between curves, volumes of revolution, and improper integrals.

定积分是微积分的基石，表示面积、体积和总变化量等量的累积。在 IB 数学 HL 课程中，学生通过黎曼和、微积分基本定理、多种积分技巧以及丰富的应用（包括曲线间的面积、旋转体体积和反常积分）来探索定积分。

1. Definition and Riemann Sums | 定义与黎曼和

The definite integral of a function f on [a, b] is defined as the limit of Riemann sums: ∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx, where Δx = (b−a)/n and x_i* is any sample point in the i‑th subinterval. Common choices include left endpoints, right endpoints, and midpoints. When this limit exists and is independent of the choice of sample points, f is integrable on [a, b].

函数 f 在 [a, b] 上的定积分定义为黎曼和的极限：∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx，其中 Δx = (b−a)/n，x_i* 是第 i 个子区间内的任意样本点。常用的取法包括左端点、右端点和中点。当该极限存在且与样本点取法无关时，f 在 [a, b] 上可积。

negative values, the integral represents the net signed area, counting regions below the x‑ A simple example is f(x) = x² on [0,1]; using right endpoints, the Riemann sum is Σ (i/n)² ⋅ (1/n), whose limit as n→∞ is 1/3, showing ∫₀¹ x² dx = 1/3.

从几何上看，当 f(x) ≥ 0 时，定积分表示曲线 y = f(x) 下方的面积。如果 f 取负值，积分表示有向面积的代数和，x 轴下方的区域计为负。一个简单例子是 f(x) = x² 在 [0,1] 上；取右端点，黎曼和为 Σ (i/n)² ⋅ (1/n)，当 n→∞ 时极限为 1/3，即得 ∫₀¹ x² dx = 1/3。

2. Fundamental Theorem of Calculus | 微积分基本定理

Calculus (FTC) has two parts. Part 1 states that if f is continuous on [a, b] and F(x) = ∫_a^x f(t) dt, then F' ). Part 2 (often used for computation) tells us that if F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). This bridges differentiation and integration, allowing definite integrals to be evaluated without Riemann sums.

微积分基本定理（FTC）包含两部分。第一部分指出，若 f 在 [a, b] 上连续，且 F(x) = ∫_a^x f(t) dt，则 F'(x) = f(x)。第二部分（常用于计算）告诉我们，如果 F 是 f 的任何一个原函数，那么 ∫_a^b f(x) dx = F(b) − F(a)。这架起了微分与积分之间的桥梁，使得不必使用黎曼和就能计算定积分。

For example, to evaluate ∫₁³ 2x dx, we note that an antiderivative of 2x is x². Thus ∫₁³ 2x dx = [x²]₁³ continuous on the closed interval; otherwise, the FTC may not apply directly, as seen later with improper integrals.

例如，要计算 ∫₁³ 2x dx，我们注意到 2x 的一个原函数是 x²，所以 ∫₁³ 2x dx = [x²]₁³ = 9 − 1 = 8。一个关键要求是 f 在闭区间上连续；否则 FTC 可能不能直接应用，这将在后续反常积分中见到。

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3. Basic Properties of Definite Integrals | 定积分的基本性质

Linearity: ∫_a^b [c f(x) + d g(x)] dx = c ∫_a^b f(x) dx + d ∫_a^b g(x) dx for constants c, d.

线性性：对于常数 c, d，有 ∫_a^b [c f(x) + d g(x)] dx = c ∫_a^b f(x) dx + d ∫_a^b g(x) dx。

Interval additivity: ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx for any c ∈ [a, b].

区间可加性：对任意 c ∈ [a, b]，有 ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx。

Reversal of limits: ∫_a^b f(x) dx = −∫_b^a f(x) dx.

积分限互换：∫_a^b f(x) dx = −∫_b^a f(x) dx。

Comparison: If f(x) ≤ g(x) on [a, b], then ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx. In particular, if m ≤ f(x) ≤ M, then m(b−a) ≤ ∫_a^b f(x) dx ≤ M(b−a).

比较性质：若在 [a, b] 上 f(x) ≤ g(x)，则 ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx。特别地，若 m ≤ f(x) ≤ M，则 m(b−a) ≤ ∫_a^b f(x) dx ≤ M(b−a)。

Definite integral of an absolute value: ∣∫_a^b f(x) dx∣ ≤ ∫_a^b ∣f(x)∣ dx.

绝对值积分不等式：∣∫_a^b f(x) dx∣ ≤ ∫_a^b ∣f(x)∣ dx。

These properties often simplify computations and allow estimates without evaluating the integral completely.

这些性质经常能简化计算，并允许在不完全计算积分的情况下进行估值。

4. Evaluating Definite Integrals with Antiderivatives | 利用原函数计算定积分

nd lower limits. The constant of integration cancels, so it can be omitted. For example, ∫₀^π sin x dx = [−cos x]₀^π = (−cos π) − (−cos 0) = (1) − (−1) = 2.

利用 FTC 第二部分，步骤很简单：找出 f(x) 的一个原函数 F(x)，然后代入上下限。积分常数会抵消，因此可以省略。例如，∫₀^π sin x dx = [−cos x]₀^π = (−cos π) − (−cos 0) = (1) − (−1) = 2。

nction that does not require substitution, we can still integrate directly. For example, ∫₀¹ e^3x dx = [ (1/3)e^3x ]₀¹ ith chain-rule factors; a common mistake is forgetting the divisor when the inner function is linear.

当被积函数是不需要换元的复合函数时，也可以直接积分。例如，∫₀¹ e^3x dx = [ (1/3)e^3x ]₀¹ = (1/3)e³ − (1/3)。必须注意链式法则的系数；常见的错误是在内层函数为线性时忘记除以系数。

5. Substitution in Definite Integrals | 定积分的换元积分

Integration by substitution, or u‑ od is to change the limits of integration together with the variable. Let u = g(x), du = g'(x) dx. When x = a, u = g(a); when x = b, u = g(b). Then ∫_a^b f(g(x)) g'(x) dx = ∫_g(a)^g(b) f(u) du.

换元积分法（u-代换）在处理定积分时有两种做法。更稳妥的方法是连同变量一起变换积分限。令 u = g(x)，du = g'(x) dx。当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。于是 ∫_a^b f(g(x)) g'(x) dx = ∫_g(a)^g(b) f(u) du。

For instance, evaluate ∫₀^√π x, so x dx = (1/2) du. When x = 0, u = 0; when x = √π, u = π. The integral becomes (1/2) ∫₀^π cos u du = (1/2) [sin u]₀^π = 0. It is crucial not to forget changing the limits – terms of u, then substitute back to x and use the original limits, but this is more error‑prone.

例如，计算 ∫₀^√π x cos(x²) dx。令 u = x²，du = 2x dx，因此 x dx = (1/2) du。当 x = 0 时，u = 0；当 x = √π 时，u = π。积分变为 (1/2) ∫₀^π cos u du = (1/2) [sin u]₀^π = 0。关键是不能忘记变换积分限——另一种做法是求出关于 u 的原函数，再代回 x 并代入原积分限，但这样更容易出错。

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