The Fundamental Theorem of Calculus | 微积分基本定理

📚 The Fundamental Theorem of Calculus | 微积分基本定理

The Fundamental Theorem of Calculus (FTC) is a central result that bridges two major branches of calculus: differentiation and integration. It reveals that these inverse operations are deeply connected, allowing us to evaluate definite integrals by finding antiderivatives and to understand how accumulation functions change. Mastering the FTC is essential for any student of IB Mathematics, as it underpins the treatment of area, total change, and the analysis of functions defined by integrals.

微积分基本定理（FTC）是连接微分和积分两大分支的核心结论。它揭示了这两种互逆运算之间的深刻联系，使我们能够通过寻找反导数来计算定积分，并理解累积函数如何变化。掌握微积分基本定理对每一位IB数学学生都至关重要，因为它是处理面积、总变化以及分析由积分定义的函数的基础。

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1. Introduction | 引言

The FTC consists of two parts. Part 1 states that if we define a function as the integral of a continuous function from a constant to a variable upper limit, then the derivative of this new function is the original function. Part 2 states that the definite integral of a function over an interval can be computed by evaluating any of its antiderivatives at the endpoints. Together, they unify the concept of area with the concept of instantaneous rate of change.

FTC包含两部分。第一部分指出，如果我们把一个函数定义为某个连续函数从常数到可变上限的积分，那么这个新函数的导数就是原来的函数。第二部分指出，函数在一个区间上的定积分可以通过计算其任意一个反导数在端点处的值来求得。两者结合，将面积的概念与瞬时变化率的概念统一了起来。

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2. Definite Integrals and Area | 定积分与面积

A definite integral ∫_a^b f(x) dx represents the signed area between the graph of f and the x‑ the integral gives the area of the region. If f changes sign, the integral computes the net area (area above the x‑ t definition using Riemann sums is fundamental, but the FTC gives us a practical shortcut.

定积分 ∫_a^b f(x) dx 表示从 x = a 到 x = b 函数 f 的图像与 x 轴之间的带符号的面积。如果 f 为正，积分给出的就是该区域的面积。如果 f 变号，积分计算的是净面积（x 轴上方面积减去下方面积）。用黎曼和定义的极限是基础，但FTC为我们提供了一条实用的捷径。

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3. The First Fundamental Theorem (FTC1) | 第一基本定理 (FTC1)

If f is continuous on [a, b] and we define a function F by F(x) = ∫_a^x f(t) dt for x ∈ ivative of an integral with respect to its upper limit is the integrand evaluated at that limit. This shows that integration and differentiation are inverse processes.

如果 f 在 [a, b] 上连续，并且我们定义函数 F(x) = ∫_a^x f(t) dt，x ∈ [a, b]，那么 F 在 (a, b) 上可导，且 F′(x) = f(x)。换句话说，对上限求导所得的导数值就是被积函数在该上限处的值。这表明积分和微分是互逆的过程。

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4. The Second Fundamental Theorem (FTC2) | 第二基本定理 (FTC2)

f is continuous on [a, b] and F is any antiderivative of f (that is, F′(x) = f(x)), then ∫_a^b owerful method for evaluating definite integrals: find an antiderivative, plug in the endpoints, and subtract. The result is independent of which antiderivative is chosen because the constant of integration cancels out.

如果 f 在 [a, b] 上连续，且 F 是 f 的任意一个反导数（即 F′(x) = f(x)），那么 ∫_a^b f(x) dx = F(b) − F(a)。这为计算定积分提供了一个强有力的方法：找到反导数，代入端点，然后相减。结果与选取哪一个反导数无关，因为积分常数会相互抵消。

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5. Proof Sketch of FTC1 | FTC1 的证明概要

ence quotient (F(x+h) − F(x))/h. Using properties of definite integrals, F(x+h) − F(x) = ∫_x^x+h and continuity of f gives f(c) → f(x). Hence the limit of the difference quotient is f(x).

要证明 F′(x) = f(x)，我们考虑差商 (F(x+h) − F(x))/h。利用定积分的性质，F(x+h) − F(x) = ∫_x^x+h f(t) dt。根据积分中值定理，存在某个 c ∈ [x, x+h] 使得该积分等于 f(c)·h。当 h → 0 时，c → x，由 f 的连续性得 f(c) → f(x)。因此差商的极限就是 f(x)。

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6. Proof Sketch of FTC2 | FTC2 的证明概要

Define G(x) = ∫_a^x or some constant C. Evaluating F(b) − F(a) gives (G(b) + C) − (G(a) + C) = G(b) − G(a) = ∫_a^b f(t) dt − 0 = ∫_a^b f(x) dx. The proof elegantly relies on FTC1 and the fact that two antiderivatives differ by a constant.

定义 G(x) = ∫_a^x f(t) dt。由 FTC1，G′(x) = f(x)，因此 G 是 f 的一个反导数。若 F 是任意另一个反导数，则 F(x) = G(x) + C，其中 C 为常数。计算 F(b) − F(a) 得 (G(b) + C) − (G(a) + C) = G(b) − G(a) = ∫_a^b f(t) dt − 0 = ∫_a^b f(x) dx。该证明精巧地依赖于 FTC1 以及两个反导数相差一个常数的事实。

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7. Geometric Interpretation | 几何解释

FTC1 can be visualised by thinking of F(x) as the area under f from a to x. Adding a small increment h increases the area by approximately f(x)·h (the area of a thin rectangle). Thus the rate of area growth is f(x). FTC2 states that the total change in F over [a, b] equals the accumulation of the rate f. This mirrors the concept in physics: if f is velocity, then the displacement is the integral of velocity, and the antiderivative of velocity gives the position function.

FTC1 可以这样可视化：将 F(x) 视为 f 从 a 到 x 下方的面积。增加一个小增量 h，面积大约增加了 f(x)·h（一个细长矩形的面积）。因此面积的增长率就是 f(x)。FTC2 则表明 F 在 [a, b] 上的总变化等于变化率 f 的累积。这与物理学中的概念相呼应：如果 f 是速度，那么位移就是速度的积分，而速度的反导数给出位置函数。

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8. Applications: Evaluating Definite Integrals | 应用：计算定积分

Using FTC2, we can compute ∫₀^π sin x dx = [−cos x]₀^π 1) − (−1) = 2. Here the antiderivative of sin x is −cos x. More generally, for any power xⁿ (n ≠ −1), an antiderivative is xⁿ⁺¹/(n+1). Thus ∫_a^b xⁿ dx = [xⁿ⁺¹/(n+1)]_a^b. This technique replaces limit‑of‑Riemann‑sum calculations with algebraic evaluations.

利用 FTC2，我们可以计算 ∫₀^π sin x dx = [−cos x]₀^π = (−cos π) − (−cos 0) = (1) − (−1) = 2。这里 sin x 的反导数是 −cos x。更一般地，对于任意幂函数 xⁿ (n ≠ −1)，其一个反导数是 xⁿ⁺¹/(n+1)。因此 ∫_a^b xⁿ dx = [xⁿ⁺¹/(n+1)]_a^b。这一技巧将黎曼和极限的计算转化为代数求值。

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9. Applications: Accumulation Functions | 应用：累积函数

FTC1 is used when a function is defined by an integral, such as g(x) = ∫₁^x o find an elementary antiderivative. When the upper limit is a function of x, say g(x) = ∫_a^u(x) gives g′(x) = f(u(x))·u′(x). This is vital in IB problems involving derivative of integral‑defined functions.

当一个函数由积分定义时，例如 g(x) = ∫₁^x cos(t²) dt，可利用 FTC1 直接写出 g′(x) = cos(x²)，而无需找出初等反导数。当积分上限是 x 的函数，如 g(x) = ∫_a^u(x) f(t) dt，链式法则给出 g′(x) = f(u(x))·u′(x)。这在涉及积分定义函数求导的IB问题中至关重要。

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10. Common Mistakes and Pitfalls | 常见错误与陷阱

One frequent error is forgetting to change the sign when swapping limits: ∫_a^b f = − ∫_b^a f. Another is misapplying the variable: in ∫_a^x f(t) dt the variable of integration is t, not x, so the derivative with respect to x is f(x), not f(t). Also, the FTC requires f to be continuous on the closed interval; discontinuities can lead to incorrect results if not carefully handled. Finally, do not forget the chain rule when the upper limit is a function of x.

一个常见错误是交换积分限时忘记变号：∫_a^b f = − ∫_b^a f。另一个是误用变量：在 ∫_a^x f(t) dt 中，积分变量是 t，而不是 x，所以对 x 求导结果是 f(x)，而非 f(t)。此外，FTC 要求 f 在闭区间上连续；若有间断点且未妥善处理，可能导致错误结果。最后，当上限是 x 的函数时，切勿遗漏链式法则。

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11. Worked Examples | 典型例题

Example 1: Evaluate ∫₁⁴ x + 3) dx. An antiderivative is x² + 3x. Then F(4) − F(1) = (16+12) − (1+3) = 28 − 4 = 24.

例1：计算 ∫₁⁴ (2x + 3) dx。一个反导数是 x² + 3x。则 F(4) − F(1) = (16+12) − (1+3) = 28 − 4 = 24。

Example 2: If h(x) = ∫₂^x² n t dt, find h′(x). By FTC1 and the chain rule, h′(x) = ln(x²) · 2x = 4x ln x (for x > 0).

例2：若 h(x) = ∫₂^x² ln t dt，求 h′(x)。由 FTC1 和链式法则，h′(x) = ln(x²) · 2x = 4x ln x（x > 0）。

Example 3: Find the area under f(x) = e^2x from x = 0 to x = 1. ∫₀¹ e^2x dx = [½ e^2x]₀¹ = ½ e² − ½.

例3：求 f(x) = e^2x 从 x = 0 到 x = 1 下方的面积。∫₀¹ e^2x dx = [½ e^2x]₀¹ = ½ e² − ½。

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12. Summary | 总结

The Fundamental Theorem of Calculus unites the two pillars of calculus. FTC1 tells us that differentiation undoes integration, while FTC2 tells us that integration via antiderivatives undoes differentiation. Mastery of these ideas enables efficient calculation of definite integrals, a deep geometric understanding of accumulation, and the ability to handle functions expressed as integrals. In IB examinations, the FTC appears in a wide range of contexts, from simple evaluations to finding derivatives of variable‑limit integrals.

微积分基本定理将微积分的两大支柱统一起来。FTC1 告诉我们微分可以撤销积分，而 FTC2 告诉我们通过反导数进行积分可以撤销微分。掌握这些思想，能够高效地计算定积分，深刻理解累积的几何意义，并处理由积分表达的函数。在IB考试中，FTC 以各种形式出现，从简单的定积分计算到可变限积分的求导，无所不包。

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