The Definite Integral and Area | 定积分与面积

📚 The Definite Integral and Area | 定积分与面积

The definite integral is one of the central concepts in IB Mathematics, providing a powerful tool not only for summing infinitesimal quantities but also for calculating the area under a curve. This article explores how the limit of a Riemann sum leads to the definite integral, how to evaluate it using the Fundamental Theorem of Calculus, and how to interpret the result as a signed area. We will examine areas between curves, integration with respect to the y‑axis, and parametric curves, linking rigorous mathematical definitions to practical geometric meaning.

定积分是 IB 数学的核心概念之一，它既是无穷小量求和的强大工具，也是计算曲线下面积的基本方法。本文将探讨黎曼和的极限如何导出定积分，如何利用微积分基本定理进行计算，以及如何将结果解释为带符号的面积。我们还将研究两曲线之间的面积、关于 y 轴的积分和参数曲线的面积，将严谨的数学定义与实际的几何意义联系起来。

1. Definition of the Definite Integral | 定积分的定义

the limit of a Riemann sum: the signed area of the region bounded by the graph of f, the x‑axis, and the vertical lines x = a and x = b. Notation: ∫_a^b f(x) dx. The symbol ∫ is an elongated S for “sum”; f(x) is the integrand; a and b are the lower and upper limits of integration.

连续函数 f 在区间 [a, b] 上的定积分定义为黎曼和的极限：由 f 的图形、x 轴以及直线 x = a 和 x = b 所围区域的带符号面积。记作 ∫_a^b f(x) dx。符号 ∫ 是拉长的 S，表示“求和”；f(x) 是被积函数；a 和 b 分别是积分下限和上限。

∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i^*) Δx

Here Δx = (b − a)/n and x_i^* is a sample point in the i‑th subinterval. The limit exists if f is continuous or piecewise continuous on [a, b]; then we say f is integrable.

其中 Δx = (b − a)/n，x_i^* 是第 i 个子区间内的样本点。若 f 在 [a, b] 上连续或分段连续，则该极限存在，此时称 f 可积。

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2. Riemann Sums and the Limit | 黎曼和与极限

en at the left endpoint, right endpoint, or midpoint. The approximation improves as n → ∞.

黎曼和通过将区域划分为 n 个等宽的细长方形矩形 (宽度为 Δx) 来逼近曲线下的面积。每个矩形的高可在左端点、右端点或中点处取值。当 n → ∞ 时，近似值趋于精确值。

Common Riemann sums include:

常见的黎曼和包括：

• Left sum: L_n = Σ_i=0ⁿ⁻¹ f(x_i) Δx — 左和
• Right sum: R_n = Σ_i=1ⁿ f(x_i) Δx — 右和
• Midpoint sum: M_n = Σ_i=1ⁿ f((x_i−1+x_i)/2) Δx — 中点和

For IB, you may be required to write a Riemann sum using sigma notation and interpret its limit as a definite integral. Practice translating a Riemann sum into an integral and vice‑versa.

在 IB 中，你可能需要利用求和符号写出黎曼和，并将其解释为定积分。练习在黎曼和与积分之间相互转换。

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

em of Calculus (FTC) links differentiation and integration. Part 1 states that if F(x) = ∫_a^x f(t) dt, then F' es a practical way to evaluate definite integrals: if F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a).

微积分基本定理 (FTC) 将微分和积分联系起来。第一部分指出，若 F(x) = ∫_a^x f(t) dt，则 F'(x) = f(x)。第二部分给出计算定积分的实用方法：若 F 是 f 的任一原函数，则 ∫_a^b f(x) dx = F(b) − F(a)。

∫_a^b f(x) dx = [F(x)]_a^b = F(b) − F(a)

The FTC guarantees that area calculations become straightforward once you find an antiderivative. Always check continuity on [a, b] before applying the FTC.

FTC 保证，一旦找到原函数，面积计算就变得直接。在应用 FTC 之前，务必检查 f 在 [a, b] 上是否连续。

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4. Basic Rules of Definite Integration | 定积分的基本法则

Definite integrals obey linearity rules and interval properties. Key properties:

定积分满足线性性质和区间性质。关键性质：

• ∫_a^b k f(x) dx = k ∫_a^b f(x) dx — 常数因子可提出
• ∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx — 和差法则
• ∫_a^b f(x) dx = −∫_b^a f(x) dx — 交换上下限变号
• ∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx — 区间可加性
• If f(x) ≥ 0 on [a, b], then ∫_a^b f(x) dx ≥ 0 — 保号性

These rules allow you to split complicated integrals into simpler parts or to reverse bounds when necessary. Always be careful with absolute values or piecewise functions.

这些法则允许你将复杂积分拆分成较简单的部分，或在需要时颠倒上下限。处理绝对值或分段函数时要格外小心。

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5. Area Between a Curve and the x‑axis | 曲线与 x 轴之间的面积

The geometric area between the curve y = f(x) and the x‑axis from x = a to x = b is given by ∫_a^b |f(x)| dx, because the definite integral itself gives signed area. Where f(x) < 0, the integral counts as negative area.

曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的几何面积由 ∫_a^b |f(x)| dx 给出，因为定积分本身给出的是带符号的面积。当 f(x) < 0 时，积分值为负面积。

To calculate total area, identify where f crosses the x‑axis (find roots), split the interval, integrate, and add absolute values:

要计算总面积，先确定 f 与 x 轴的交点 (求根)，分割区间，分别积分再取绝对值相加：

Total area = ∫_a^c (−f(x)) dx + ∫_c^b f(x) dx

if f(x) < 0 on [a, c] and f(x) > 0 on [c, b].

例如，若 f(x) 在 [a, c] 上为负，在 [c, b] 上为正，则总面积 = ∫_a^c (−f(x)) dx + ∫_c^b f(x) dx。

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6. Area Between Two Curves | 两曲线之间的面积

er) and y = g(x) (lower) intersect at x = a and x = b, the area enclosed between them is ∫_a^b ) throughout the interval. If the curves cross, split the integral at intersection points.

若两条曲线 y = f(x) (上方) 和 y = g(x) (下方) 在 x = a 与 x = b 处相交，则它们之间的面积为 ∫_a^b [f(x) − g(x)] dx，前提是区间内始终有 f(x) ≥ g(x)。如果曲线相交，应在交点处分割积分。

alues to determine which is upper; (3) Set up integral ∫ (upper − lower) dx; (4) Evaluate.

步骤：(1) 解方程 f(x) = g(x) 求交点；(2) 画图或代入测试值确定上方曲线；(3) 建立积分 ∫ (上 − 下) dx；(4) 计算。

Scenario	Integral setup
Curves in terms of x	∫ab (top − bottom) dx
Curves in terms of y	∫cd (right − left) dy

pressed as x = g(y) and x = h(y), and you integrate with respect to y from y = c to y = d.

在关于 y 积分的情况下，函数必须写成 x = g(y) 和 x = h(y)，并从 y = c 到 y = d 进行积分。

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7. Handling Negative Areas and Absolute Value | 处理负面积与绝对值

When a function dips below the x‑axis, the integral itself becomes negative. To obtain total geometric area, you must use absolute values or integrate −f(x) on the negative portions. The same principle applies when finding the area between two curves that cross: use |f(x) − g(x)| or split the domain.

当函数下降到 x 轴以下时，积分值本身为负。要得到总的几何面积，必须使用绝对值，或在负值部分对 −f(x) 积分。同样的原则适用于求两条相交曲线之间的面积：使用 |f(x) − g(x)| 或分割定义域。

Example: Find the area bounded by y = x³ − 4x and the x‑ integrate in sections, taking the negative of the integral where the function is negative.

示例：求 y = x³ − 4x 与 x 轴所围面积。先解 x³ − 4x = 0 → x = −2, 0, 2。然后分段积分，在函数为负的区间取积分的相反数。

Area = ∫₋₂⁰ (x³ − 4x) dx + ∫₀² (−(x³ − 4x)) dx

Check your answer using a calculator: the total area is 8 square units.

用计算器检查：总面积为 8 平方单位。

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8. Integration with Respect to y | 关于 y 的积分求面积

rves are defined x = f(y). The area between x = f(y) and x = g(y) from y = c to y = d is ∫_c^d (right − left) dy.

有时将自变量视为 y 会更简单，尤其对于侧向抛物线或曲线以 x = f(y) 形式给出时。从 y = c 到 y = d 之间 x = f(y) 与 x = g(y) 的面积为 ∫_c^d (右 − 左) dy。

he line x = 4 can be found by integrating with respect to y from y = −2 to y = 2: Area = ∫₋₂² (4 − y²) dy.

例如，曲线 x = y² 与直线 x = 4 所围面积可通过对 y 积分求出：y 从 −2 到 2，面积 = ∫₋₂² (4 − y²) dy。

Always sketch the region to correctly identify boundaries and the right/left functions. In the dy‑setup, the roles of “top/bottom” become “right/left”.

始终画出区域草图，以正确识别边界和左/右函数。在 dy 设定中，“上/下”变成“右/左”。

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9. Areas of Parametrically Defined Curves | 参数方程所定义曲线的面积

ea under the curve from t = α to t = β (where the curve moves monotonically in x) is A = ∫_α^β y(t) x' is derived from substitution in the standard area formula: A = ∫ y dx = ∫ y(t) (dx/dt) dt.

对于由参数方程 x = x(t), y = y(t) 给出的曲线，从 t = α 到 t = β (且曲线在 x 方向单调移动) 的曲线下方面积为 A = ∫_α^β y(t) x'(t) dt。这是由代入标准面积公式 A = ∫ y dx = ∫ y(t) (dx/dt) dt 推导而来的。

ch: A = ½ ∫ (x dy − y dx), but IB typically focuses on the simple parametric area formula.

如果曲线并非单调，则需分成 x 严格递增或递减的区间，并相应地调整符号。对于闭合区域，面积可通过格林定理型方法计算：A = ½ ∫ (x dy − y dx)，但 IB 通常关注简单的参数面积公式。

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10. Application Problems: Finding Area | 应用问题：求面积

IB exam questions often embed area problems in real‑world contexts, such as the area of a plot of land, the cross‑sectional area of a tunnel, or the area of a region bounded by speed‑ ay also combine area with velocity and displacement: ∫ v(t) dt gives displacement, while ∫ |v(t)| dt gives total distance travelled (area between the velocity curve and the t‑axis).

IB 考题常将面积问题嵌入实际情境，例如地块面积、隧道横截面积，或速度‑时间图所围区域。还可能将面积与速度和位移结合：∫ v(t) dt 给出位移，而 ∫ |v(t)| dt 给出总行驶距离 (速度曲线与 t 轴之间的面积)。

Approach: identify what the area represents; sketch the region; determine the appropriate integration variable (x, y, or t); set up the integral with correct limits and integrand; compute exactly or approximate using GDC.

方法：确定面积代表的意义；画出区域；确定合适的积分变量 (x、y 或 t)；建立带有正确上下限和被积函数的积分；精确计算或用 GDC 近似计算。

Non‑standard bounds: sometimes the region is bounded by vertical/horizontal lines or asymptotes; use improper integrals if needed, but IB AA HL focuses on proper integrals, whereas some HL option topics may include improper integrals.

非标准边界：有时区域由垂直/水平线或渐近线界定；若需要可使用反常积分，但 IB AA 主要关注正常积分，而部分 HL 选考内容可能包含反常积分。

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11. GDC Techniques for Definite Integrals | GDC 技巧求定积分

Your graphical display calculator (GDC) can evaluate definite integrals numerically. Common steps: enter the function; use the ‘integrate’ command (often ‘fnInt’ or a specific calculus menu); input lower limit, upper limit, and variable. The GDC gives an approximate decimal value, which is usually sufficient unless exact answers are required.

图形计算器 (GDC) 可对定积分进行数值计算。常见步骤：输入函数；使用“积分”命令 (常为 ‘fnInt’ 或特定微积分菜单)；输入下限、上限和变量。GDC 给出近似的十进制数值，除非要求精确答案，这通常足够。

To find the area between curves with GDC: graph both functions; use the ‘intersect’ feature to find bounds; then integrate the difference function numerically. Many GDC models also allow shading the region and computing its area directly via the Analyze Graph tool.

用 GDC 求两曲线之间的面积：画出两个函数图像；使用“交点”功能找边界；然后数值积分差值函数。许多 GDC 型号还支持直接通过分析图形工具着色区域并计算面积。

Remember: the calculator is a powerful check, but you must still show analytical steps in exams unless the question explicitly asks for a GDC‑only solution.

记住：计算器是强大的检验工具，但在考试中仍需写出解析步骤，除非题目明确要求仅用 GDC 求解。

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12. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Pitfall 1: Forgetting that the definite integral yields signed area. Always check for parts below the x‑axis when computing total geometric area. Use absolute values or split the integral.

误区 1：忘记定积分给出的是带符号面积。计算总几何面积时务必检查 x 轴下方的部分，使用绝对值或拆分积分。

Pitfall 2: Mixing up “upper minus lower” order in area between curves. The integrand must be (top − bottom) or (right − left); otherwise the result will be negative but still correct in magnitude – careful.

误区 2：在曲线间面积中混淆“上减下”的顺序。被积函数必须是 (上 − 下) 或 (右 − 左)，否则结果为负但绝对值仍正确——需要注意。

Pitfall 3: Not finding all intersection points. Sketch graphs, solve equations carefully, and consider domain restrictions.

误区 3：没有找到所有交点。绘制草图，仔细解方程，并考虑定义域限制。

Pitfall 4: Incorrect use of the FTC, such as differentiating instead of integrating, or misapplying the chain rule when the upper limit is a function of x.

误区 4：错误应用 FTC，例如应积分却求导了，或当积分上限是 x 的函数时未正确应用链式法则。

Exam tip: Always write the integral setup before evaluating. Include dx or dy. Label your antiderivative clearly and show the substitution of bounds. If using GDC, state which function you integrated. A well‑presented solution earns method marks even if the final numerical answer is slightly off.

考试技巧：计算前务必写出积分设置。包含 dx 或 dy。清晰地写出原函数并展示代入上下限的步骤。若使用 GDC，说明对哪个函数进行了积分。即使最终数值答案略有偏差，良好的解题过程也能获得方法分。

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