📚 The Fundamental Theorem of Calculus and Its Applications | 微积分基本定理及其应用
The Fundamental Theorem of Calculus (FTC) is the bridge between differentiation and integration, two pillars of calculus. It states that differentiation and integration are inverse processes, and it provides a powerful method for evaluating definite integrals without resorting to limit definitions.
微积分基本定理是连接微分与积分这两大微积分支柱的桥梁。它指出微分与积分互为逆运算，并为我们提供了一种无需依赖极限定义即可计算定积分的强大方法。
1. The First Fundamental Theorem | 第一基本定理
The First Fundamental Theorem of Calculus states that if f is continuous on [a, b] and F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). This theorem transforms the problem of computing areas under curves into the problem of finding antiderivatives.
微积分第一基本定理指出：若 f 在 [a, b] 上连续，且 F 是 f 的一个原函数，则 f 从 a 到 b 的定积分等于 F(b) − F(a)。这一定理将计算曲线下面积的问题转化为寻找原函数的问题。
∫ₐᵇ f(x) dx = F(b) − F(a)
For example, to evaluate ∫₀¹ x² dx, we find an antiderivative F(x) = x³/3. Then F(1) − F(0) = 1/3 − 0 = 1/3.
例如，计算 ∫₀¹ x² dx 时，我们找到原函数 F(x) = x³⁄₃。则 F(1) − F(0) = 1⁄₃ − 0 = 1⁄₃。
2. The Second Fundamental Theorem | 第二基本定理
The Second Fundamental Theorem of Calculus states that if f is continuous on [a, b], then the function g(x) = ∫ₐˣ f(t) dt is differentiable on (a, b), and g′(x) = f(x). This theorem establishes that every continuous function has an antiderivative, namely the accumulation function.
微积分第二基本定理指出：若 f 在 [a, b] 上连续，则函数 g(x) = ∫ₐˣ f(t) dt 在 (a, b) 上可导，且 g′(x) = f(x)。该定理确立了每个连续函数都存在原函数，即累积函数本身。
This theorem is particularly useful for differentiating integrals with variable upper limits. For instance, if g(x) = ∫₁ˣ (t² + 1) dt, then g′(x) = x² + 1.
这一定理在求变上限积分的导数时尤为有用。例如，若 g(x) = ∫₁ˣ (t² + 1) dt，则 g′(x) = x² + 1。
3. Linking Differentiation and Integration | 联系微分与积分
Before the FTC, areas were computed using Riemann sums — the limit of rectangular approximations. The FTC reveals that areas are naturally linked to slopes: the area under a velocity-time graph gives displacement, while the slope of a displacement-time graph gives velocity.
在微积分基本定理出现之前，面积是通过黎曼和（矩形逼近的极限）来计算的。基本定理揭示了面积与斜率之间的天然联系：速度-时间图像下的面积给出位移，而位移-时间图像的斜率给出速度。
The two parts of the FTC together express a deep symmetry: differentiating an integral returns the original function, and integrating a derivative returns the original function up to a constant.
基本定理的两部分共同表达了一种深刻的对称性：对积分求导可还原原函数，而对导数积分则可还原原函数（至多相差一个常数）。
d/dx [∫ₐˣ f(t) dt] = f(x) and ∫ₐᵇ F′(x) dx = F(b) − F(a)
4. Evaluating Definite Integrals | 计算定积分
The most direct application of the FTC is evaluating definite integrals. The procedure is: find an antiderivative of the integrand, then substitute the limits of integration.
微积分基本定理最直接的应用是计算定积分。步骤为：先找到被积函数的一个原函数，再代入积分上下限。
Consider ∫₀^π sin x dx. An antiderivative is −cos x. Therefore, ∫₀^π sin x dx = −cos π − (−cos 0) = −(−1) − (−1) = 1 + 1 = 2.
考虑 ∫₀^π sin x dx。一个原函数是 −cos x。因此，∫₀^π sin x dx = −cos π − (−cos 0) = −(−1) − (−1) = 1 + 1 = 2。
When evaluating definite integrals, always compute F(b) − F(a), not F(a) − F(b). A common exam mistake is reversing the order of substitution.
计算定积分时，务必计算 F(b) − F(a)，而非 F(a) − F(b)。考试中常见的错误是颠倒代入顺序。
5. Applying the Chain Rule to the FTC | 链式法则与基本定理结合
When the upper limit of integration is itself a function of x, we must apply the chain rule. If g(x) = ∫ₐ^{u(x)} f(t) dt, then g′(x) = f(u(x)) · u′(x).
当积分上限本身是 x 的函数时，我们必须运用链式法则。若 g(x) = ∫ₐ^{u(x)} f(t) dt，则 g′(x) = f(u(x)) · u′(x)。
For example, if g(x) = ∫₀^{x²} cos(t) dt, then g′(x) = cos(x²) · 2x = 2x cos(x²). This technique frequently appears in A-level exam questions.
例如，若 g(x) = ∫₀^{x²} cos(t) dt，则 g′(x) = cos(x²) · 2x = 2x cos(x²)。这一技巧在 A-level 考试题中频繁出现。
A useful simplification: ∫ₐ^{u(x)} f(t) dt = F(u(x)) − F(a), so differentiation gives F′(u(x)) · u′(x) = f(u(x)) · u′(x). Note that the lower limit a contributes nothing, since F(a) is a constant.
一个有用的化简方式是：∫ₐ^{u(x)} f(t) dt = F(u(x)) − F(a)，因此求导得到 F′(u(x)) · u′(x) = f(u(x)) · u′(x)。注意下限 a 对结果没有贡献，因为 F(a) 是常数。
6. Finding Areas under Curves | 求曲线下方面积
One of the most important geometric applications of the FTC is computing areas. The area bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is given by ∫ₐᵇ |f(x)| dx, where the absolute value ensures that the area is always positive.
微积分基本定理最重要的几何应用之一是计算面积。由曲线 y = f(x)、x 轴以及直线 x = a、x = b 所围成的面积由 ∫ₐᵇ |f(x)| dx 给出，其中绝对值确保面积始终为正。
If f(x) ≥ 0 on [a, b], the area is simply ∫ₐᵇ f(x) dx. However, if the curve dips below the x-axis, the definite integral gives a negative value; in such cases we must split the integral at the roots and take absolute values of each segment.
若 f(x) ≥ 0 在 [a, b] 上成立，则面积直接为 ∫ₐᵇ f(x) dx。然而，若曲线降到 x 轴下方，定积分将给出负值；此时需要在根处分割积分区间并对各段取绝对值。
For instance, the area enclosed by y = x² − 1 and the x-axis from x = 0 to x = 2 is computed in two parts: ∫₀¹ (1 − x²) dx + ∫₁² (x² − 1) dx = 2/3 + 4/3 = 2.
例如，由 y = x² − 1 与 x 轴从 x = 0 到 x = 2 所围成的面积需分两部分计算：∫₀¹ (1 − x²) dx + ∫₁² (x² − 1) dx = 2⁄₃ + 4⁄₃ = 2。
7. Area between Curves | 两曲线之间的面积
To find the area between two curves y = f(x) and y = g(x), where f(x) ≥ g(x) on [a, b], the formula is ∫ₐᵇ [f(x) − g(x)] dx. This can be seen as the integral of the “top minus bottom” function.
要求两条曲线 y = f(x) 与 y = g(x) 之间的面积，其中 f(x) ≥ g(x) 在 [a, b] 上成立，公式为 ∫ₐᵇ [f(x) − g(x)] dx。这可以理解为“上减下”函数的积分。
Example: Find the area between y = x² and y = x from x = 0 to x = 1. Since x ≥ x² on [0, 1], the area is ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
例：求 y = x² 与 y = x 在 x = 0 到 x = 1 之间的面积。因在 [0, 1] 上 x ≥ x²，故面积为 ∫₀¹ (x − x²) dx = [x²⁄₂ − x³⁄₃]₀¹ = 1⁄₂ − 1⁄₃ = 1⁄₆。
When the intersection points are not given, you must first solve f(x) = g(x) to determine the limits of integration.
当交点未给定时，必须先解方程 f(x) = g(x) 以确定积分上下限。
8. Volumes of Revolution | 旋转体体积
The FTC also enables the computation of volumes of solids generated by rotating a curve about an axis. When the region under y = f(x) from a to b is rotated about the x-axis, the volume is given by V = π ∫ₐᵇ [f(x)]² dx.
微积分基本定理还可用于计算曲线绕轴旋转所生成的三维体积。当 y = f(x) 在 a 到 b 间的区域绕 x 轴旋转时，体积为 V = π ∫ₐᵇ [f(x)]² dx。
In this formula, each thin disk of radius f(x) and thickness dx has volume π[f(x)]² dx; the FTC converts the sum of infinitely many disks into a computable integral.
在此公式中，每个半径为 f(x)、厚度为 dx 的薄圆盘的体积为 π[f(x)]² dx；基本定理将无穷多个圆盘的累加转化为可计算的积分。
For example, the volume of a sphere of radius r is obtained by rotating y = √(r² − x²) from −r to r about the x-axis: V = π ∫₋ᵣʳ (r² − x²) dx = π [r²x − x³/3]₋ᵣʳ = 4πr³/3.
例如，半径为 r 的球体体积可由 y = √(r² − x²) 从 −r 到 r 绕 x 轴旋转得到：V = π ∫₋ᵣʳ (r² − x²) dx = π [r²x − x³⁄₃]₋ᵣʳ = 4πr³⁄₃。
9. Applications in Motion: displacement, velocity, acceleration | 运动学中的应用：位移、速度、加速度
In kinematics, the FTC provides a direct link between acceleration, velocity, and displacement. If a(t) is acceleration, then v(t) = v(t₀) + ∫ₜ₀ᵗ a(u) du, and displacement is s(t) = s(t₀) + ∫ₜ₀ᵗ v(u) du.
在运动学中，微积分基本定理建立了加速度、速度和位移之间的直接联系。若 a(t) 为加速度，则 v(t) = v(t₀) + ∫ₜ₀ᵗ a(u) du，位移为 s(t) = s(t₀) + ∫ₜ₀ᵗ v(u) du。
A typical A-level problem: a particle moves with velocity v(t) = 3t² − 12t + 9 m/s. Find the displacement during the first 4 seconds and the total distance travelled.
一个典型的 A-level 问题：某质点以速度 v(t) = 3t² − 12t + 9 m/s 运动。求前 4 秒内的位移和总路程。
Displacement = ∫₀⁴ (3t² − 12t + 9) dt = [t³ − 6t² + 9t]₀⁴ = 64 − 96 + 36 = 4 m. For distance, we integrate the absolute value of velocity, noting v(t) = 3(t − 1)(t − 3), so v = 0 at t = 1 and t = 3.
位移 = ∫₀⁴ (3t² − 12t + 9) dt = [t³ − 6t² + 9t]₀⁴ = 64 − 96 + 36 = 4 米。对于路程，需对速度的绝对值积分，注意 v(t) = 3(t − 1)(t − 3)，因此在 t = 1 和 t = 3 处速度为 0。
Total distance = ∫₀¹ v dt − ∫₁³ v dt + ∫₃⁴ v dt = 4 + 4 + 4 = 12 m. The displacement is small, but the distance travelled is much larger because the particle reverses direction twice.
总路程 = ∫₀¹ v dt − ∫₁³ v dt + ∫₃⁴ v dt = 4 + 4 + 4 = 12 米。位移很小，但路程要大得多，因为质点两次改变了运动方向。
10. Techniques: substitution and integration by parts | 技巧：换元法与分部积分法
While the FTC gives the framework for evaluating integrals, many integrands require advanced techniques. Substitution (u-substitution) is used when the integrand contains a composite function. For definite integrals, the limits must also be transformed.
虽然基本定理提供了计算积分的框架，但许多被积函数需要更高级的技巧。换元法（u 代换）适用于被积函数包含复合函数的情形。对于定积分，积分上下限也必须相应变换。
For example, to evaluate ∫₀² x(x² + 1)⁵ dx, let u = x² + 1, du = 2x dx. When x = 0, u = 1; when x = 2, u = 5. Thus the integral becomes (1/2) ∫₁⁵ u⁵ du = (1/12)(5⁶ − 1).
例如，计算 ∫₀² x(x² + 1)⁵ dx 时，令 u = x² + 1，du = 2x dx。当 x = 0 时 u = 1；当 x = 2 时 u = 5。因此积分变为 (1⁄₂) ∫₁⁵ u⁵ du = (1⁄₁₂)(5⁶ − 1)。
Integration by parts is used for products of functions: ∫ u dv = uv − ∫ v du. When applying these techniques to definite integrals, be sure to evaluate all terms at both limits.
分部积分法用于处理函数乘积：∫ u dv = uv − ∫ v du。在定积分中使用这些技巧时，务必在所有项中代入上下限求值。
11. Differentiating Integrals: a unified view | 对积分求导：统一的视角
The power of the FTC lies in its ability to handle expressions that combine differentiation and integration. For instance, to find d/dx [∫ₓ² sin(t²) dt], we first swap the limits using the property ∫ₓ² = −∫²ₓ, then apply the second FTC: d/dx [−∫₂ˣ sin(t²) dt] = −sin(x²).
基本定理的优势在于它能够处理同时涉及微分与积分的表达式。例如，求 d/dx [∫ₓ² sin(t²) dt] 时，我们先利用性质 ∫ₓ² = −∫²ₓ 交换上下限，再应用第二基本定理：d/dx [−∫₂ˣ sin(t²) dt] = −sin(x²)。
More generally, d/dx [∫_{L(x)}^{U(x)} f(t) dt] = f(U(x)) · U′(x) − f(L(x)) · L′(x). This formula combines the chain rule with the FTC and handles variable lower limits as well.
更一般地，d/dx [∫_{L(x)}^{U(x)} f(t) dt] = f(U(x)) · U′(x) − f(L(x)) · L′(x)。此公式将链式法则与基本定理结合，同时处理变量下限。
12. Common Pitfalls and Exam Strategies | 常见误区与应试策略
Several mistakes commonly appear in exams. First, confusing an antiderivative with the integral — always differentiate your answer to verify. Second, forgetting the constant C when finding indefinite integrals. Third, incorrectly applying the chain rule when differentiating integrals with variable limits.
考试中有几个常见错误。第一，混淆原函数与积分——始终通过求导来验证你的答案。第二，求不定积分时忘记常数 C。第三，对变限积分求导时错误地运用链式法则。
To succeed, follow a consistent procedure: identify the type of problem, sketch graphs where needed, determine the correct formula, find antiderivatives carefully, and check whether the answer makes physical or geometric sense.
要想取得成功，应遵循一致的解题流程：识别问题类型，在需要时画出图形，确定正确的公式，仔细求原函数，并检查答案在物理或几何上是否合理。
Mastering the FTC is essential for every A-level mathematics student. It unifies seemingly unrelated topics and provides a systematic method for solving a wide variety of problems involving rates, areas, volumes, and motion.
掌握微积分基本定理是每位 A-level 数学学生必备的能力。它统一了看似无关的主题，并为解决涉及速率、面积、体积和运动的各种问题提供了系统化的方法。
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