📚 Definite Integrals: Basic Computation and Applications | 定积分的基本计算与应用
The definite integral is a cornerstone of calculus. In IB Mathematics, you are expected not only to compute definite integrals accurately but also to interpret them in geometric and real-world contexts. This article reviews the essential techniques and applications, with paired Chinese explanations to help you master the topic.
定积分是微积分的基石。在IB数学中，你不仅要能准确计算定积分，还要理解其几何与实际应用背景。本文回顾核心技巧与应用，并配有中英文对照，帮助你全面掌握这一考点。
1. Definition and Basic Properties | 定积分的定义与基本性质
The definite integral of a continuous function f(x) from a to b is defined as the limit of a Riemann sum: the signed area between the curve and the x‑axis. Key properties include linearity, additivity over intervals, and reversal of limits.
定积分是黎曼和的极限，表示曲线与x轴之间的有向面积。重要性质包括线性性、区间可加性以及上下限互换变号。
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Linearity: ∫ab [c·f(x) + d·g(x)] dx = c∫ab f(x) dx + d∫ab g(x) dx.
线性性质：∫ab [c·f(x) + d·g(x)] dx = c∫ab f(x) dx + d∫ab g(x) dx。
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Additivity: ∫ab f(x) dx + ∫bc f(x) dx = ∫ac f(x) dx.
区间可加性：∫ab f(x) dx + ∫bc f(x) dx = ∫ac f(x) dx。
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Reversal: ∫ab f(x) dx = −∫ba f(x) dx.
上下限互换：∫ab f(x) dx = −∫ba f(x) dx。
2. The Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem connects differentiation and integration. If F is an antiderivative of f, then
微积分基本定理将微分与积分联系起来。若 F 是 f 的原函数，则
∫ab f(x) dx = F(b) − F(a).
This theorem is the primary tool for evaluating definite integrals exactly. For example, ∫12 x² dx = [x³/3]12 = (8/3) − (1/3) = 7/3.
该定理是精确计算定积分的主要工具。例如 ∫12 x² dx = [x³/3]12 = 8/3 − 1/3 = 7/3。
3. Indefinite Integration and Substitution | 不定积分与换元法
Before applying the Fundamental Theorem, you need to find antiderivatives. The substitution rule reverses the chain rule. For a definite integral, remember to change the limits or convert back to the original variable.
在应用基本定理前，需要找到原函数。换元法是链式法则的逆运算。对于定积分，记得更换积分限，或最终代回原变量。
∫ab f(g(x))·g′(x) dx = ∫g(a)g(b) f(u) du.
Example: ∫01 2x·cos(x²) dx. Let u = x², du = 2x dx. When x=0, u=0; x=1, u=1. The integral becomes ∫01 cos(u) du = sin(1).
例：∫01 2x·cos(x²) dx。令 u = x²，du = 2x dx。当 x=0 时 u=0；x=1 时 u=1。积分变为 ∫01 cos(u) du = sin(1)。
4. Integration by Parts | 分部积分法
Integration by parts reverses the product rule. It is used for products of functions such as x·ex or x·sin(x). For definite integrals:
分部积分法是乘积法则的逆运算，适用于如 x·ex 或 x·sin(x) 等函数乘积的积分。定积分形式为：
∫ab u dv = [u·v]ab − ∫ab v du.
Example: ∫0π x·sin(x) dx. Let u = x, dv = sin(x) dx, then du = dx, v = −cos(x). Thus [−x·cos(x)]0π + ∫0π cos(x) dx = (−π·cos π) + [sin x]0π = π + 0 = π.
例：∫0π x·sin(x) dx。令 u = x，dv = sin(x) dx，则 du = dx，v = −cos(x)。于是 [−x·cos(x)]0π + ∫0π cos(x) dx = −π·cos π + [sin x]0π = π + 0 = π。
5. Exploiting Symmetry | 利用对称性简化计算
For definite integrals over symmetric intervals, symmetry can dramatically simplify the computation.
对于对称区间上的定积分，利用对称性可以大幅简化计算。
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If f is even, f(−x)=f(x), then ∫−aa f(x) dx = 2∫0a f(x) dx.
若 f 为偶函数，f(−x)=f(x)，则 ∫−aa f(x) dx = 2∫0a f(x) dx。
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If f is odd, f(−x)=−f(x), then ∫−aa f(x) dx = 0.
若 f 为奇函数，f(−x)=−f(x)，则 ∫−aa f(x) dx = 0。
Example: ∫−22 (x³ + x) dx = 0 because x³ and x are both odd. Meanwhile ∫−11 x⁴ dx = 2/5.
例：∫−22 (x³ + x) dx = 0，因为 x³ 和 x 都是奇函数。而 ∫−11 x⁴ dx = 2/5。
6. Area Between a Curve and the x‑axis | 曲线与x轴围成的面积
The definite integral gives the signed area, so when the curve lies below the x‑axis, the integral is negative. For the total area, split the interval at the zeros of f and take absolute values.
定积分给出的是有向面积，因此曲线位于x轴下方时积分为负。若要计算总面积，需在 f 的零点处分段并取绝对值。
Total area = ∫ab |f(x)| dx.
Example: Find the area under y = sin(x) from x=0 to x=2π. Since sin(x) is positive on (0,π) and negative on (π,2π), the total area is ∫0π sin x dx − ∫π2π sin x dx = 2 + 2 = 4.
例：求 y = sin(x) 从 x=0 到 x=2π 与x轴围成的面积。sin(x) 在 (0,π) 为正，在 (π,2π) 为负，故总面积为 ∫0π sin x dx − ∫π2π sin x dx = 2 + 2 = 4。
7. Area Between Two Curves | 两曲线之间的面积
To find the area enclosed by two curves y = f(x) and y = g(x), integrate the difference of the upper and lower functions. Determine the intersection points to set the limits.
求两条曲线 y = f(x) 与 y = g(x) 围成的面积，需要对上下函数之差积分，并通过交点确定积分限。
Area = ∫ab (f(x) − g(x)) dx, where f(x) ≥ g(x).
Example: Find the area between y = x and y = x². Intersections: x = x² → x=0,1. On [0,1], x ≥ x², so Area = ∫01 (x − x²) dx = [x²/2 − x³/3]01 = 1/2 − 1/3 = 1/6.
例：求 y = x 与 y = x² 之间的面积。交点：x = x² → x=0,1。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。
8. Volumes of Revolution | 旋转体的体积
When a region under a curve is rotated about the x‑axis, the volume is given by the disc method. For rotation about the y‑axis, use the shell method or integrate with respect to y.
当曲线下方的区域绕x轴旋转时，体积由圆盘法给出。绕y轴旋转时，可使用柱壳法或对 y 积分。
V = π ∫ab [f(x)]² dx (about the x‑axis).
Example: The volume of the solid formed by rotating y = √x, from x=0 to x=1, about the x‑axis is V = π ∫01 x dx = π/2.
例：将 y = √x（0 ≤ x ≤ 1）绕x轴旋转所得体积为 V = π ∫01 x dx = π/2。
9. Average Value of a Function | 函数的平均值
The average value of a continuous function over [a,b] is defined as the integral divided by the length of the interval. This is required in many IB applications.
连续函数在 [a,b] 上的平均值定义为积分除以区间长度。这是IB中常见的考点。
Average value = (1/(b−a)) ∫ab f(x) dx.
Example: The average value of f(x) = 3x² on [0,2] is (1/2)∫02 3x² dx = (1/2)(8) = 4.
例：f(x) = 3x² 在 [0,2] 上的平均值为 (1/2)∫02 3x² dx = (1/2)·8 = 4。
10. Applications in Kinematics | 运动学中的应用
In kinematics, if v(t) is velocity, then the definite integral of velocity gives displacement (net change in position), while the integral of speed |v(t)| gives total distance travelled.
在运动学中，若 v(t) 是速度，速度的定积分给出位移（位置净变化），而速率 |v(t)| 的积分给出总路程。
Displacement = ∫t₁t₂ v(t) dt, Distance = ∫t₁t₂ |v(t)| dt.
Example: If v(t) = t² − 4t (m/s) from t=0 to t=5, displacement = ∫05 (t² − 4t) dt = [t³/3 − 2t²]05 = 125/3 − 50 = −25/3 m. To find distance, split at v=0: t=0,4. Distance = ∫04 (−t²+4t) dt + ∫45 (t²−4t) dt = 32/3 + 7/3 = 13 m.
例：若 v(t) = t² − 4t（m/s），从 t=0 到 t=5，位移 = ∫05 (t² − 4t) dt = [t³/3 − 2t²]05 = 125/3 − 50 = −25/3 m。求路程需分段：v=0 时 t=0,4。路程 = ∫04 (−t²+4t) dt + ∫45 (t²−4t) dt = 32/3 + 7/3 = 13 m。
11. Accumulated Change and Growth Models | 累积变化与增长模型
Definite integrals are used to model total change from a rate function. For example, if a population grows at a rate r(t), the total growth over [a,b] is ∫ab r(t) dt.
定积分常用于根据变化率求总量变化。例如，若人口增长率为 r(t)，在 [a,b] 内的总增长量为 ∫ab r(t) dt。
This idea extends to economics (total revenue from marginal revenue), biology (total biomass from growth rate), and physics (charge from current). You must be able to set up and evaluate such integrals.
这一思想可推广到经济学（由边际收入求总收入）、生物学（由增长率求总生物量）和物理学（由电流求电荷）。你需要能够建立并计算这类积分。
12. Summary and Exam Tips | 总结与应试建议
Master the Fundamental Theorem, substitution, and integration by parts. Always sketch the function for area and volume questions, and remember absolute values for total area. Practice with past paper questions to build speed and accuracy.
掌握微积分基本定理、换元法和分部积分法。遇到面积和体积问题先画图，求总面积时记得取绝对值。通过真题练习提升速度与准确率。
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