📚 A - The Area Under a Curve | 曲线下的面积
The concept of the area under a curve is foundational in integral calculus. It allows us to quantify accumulation, such as distance from velocity, and provides a geometric interpretation of the definite integral. By breaking the region into infinitesimally thin slices, we sum these slices to find the total area. This idea underpins many applications in physics, economics, and beyond.
曲线下的面积是积分学的基本概念。它使我们能够量化累积量，例如由速度求距离，并为定积分提供了生动的几何解释。通过将区域划分为无穷小薄片，我们可以将这些薄片叠加求和以得到总面积。这一思想是物理、经济学及众多领域应用的基础。
1. Introduction to the Area Under a Curve | 曲线下面积简介
he limit of a sum of areas of approximating rectangles. When f(x) is positive over [a, b], this area corresponds exactly to the definite integral ∫ₐᵇ f(x) dx. The problem of squaring curved regions sparked the invention of calculus.
在数学中，曲线 y = f(x) 从 x = a 到 x = b 下方的面积定义为一系列近似矩形面积之和的极限。当 f(x) 在 [a, b] 上非负时，该面积就等同于定积分 ∫ₐᵇ f(x) dx。正是对曲线边界区域求积的追求催生了微积分。
The idea originated with the method of exhaustion used by Eudoxus and Archimedes, but it was Newton and Leibniz who recognised the inverse relationship between area accumulation and rate of change, giving us a powerful systematic tool.
这一思想源于欧多克索斯和阿基米德的穷竭法，但牛顿和莱布尼茨认识到面积累积与变化率之间的逆关系，为我们提供了强大而系统性的工具。
2. Approximating Area with Rectangles | 用矩形近似面积
of equal width Δx = (b − a)/n. On each subinterval we choose a sample point cᵢ and erect a rectangle of height f(cᵢ). The sum Σᵢ₌₁ⁿ f(cᵢ) Δx yields an estimate.
为了近似曲线 y = f(x) 在 a 与 b 之间的面积，我们把区间等分成 n 个宽度为 Δx = (b − a)/n 的子区间。在每个子区间上选取样本点 cᵢ，作出高度为 f(cᵢ) 的矩形，其总和 Σᵢ₌₁ⁿ f(cᵢ) Δx 便是一个近似值。
Common choices for cᵢ are left endpoints, right endpoints, or midpoints. For an increasing function, left Riemann sums underestimate while right sums overestimate the true area. Midpoint Riemann sums often give a closer approximation with fewer rectangles.
常见的 cᵢ 选取方式有左端点、右端点和中点。对于单调递增函数，左黎曼和会低估真实面积，而右黎曼和则会高估。中点黎曼和通常用更少的矩形就能给出更精确的近似。
3. Riemann Sums and the Limit | 黎曼和与极限
A Riemann sum for f on [a, b] is any sum of the form Rₙ = Σᵢ₌₁ⁿ f(xᵢ*) Δx, where x&# becomes finer, the Riemann sum approaches a unique value — provided f is continuous.
函数 f 在 [a, b] 上的黎曼和为 Rₙ = Σᵢ₌₁ⁿ f(xᵢ*) Δx，其中 xᵢ* 是第 i 个子区间内任意一点。当 n → ∞、分割无限变密时，只要 f 连续，黎曼和就趋近于一个唯一值。
This limiting value is the definite integral:
∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*) Δx
The notation captures both the process of summation (elongated S) and the limiting width dx.这个极限值就是定积分：
∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*) Δx
积分符号既体现了求和过程（拉长的 S），又通过 dx 反映了无穷小宽度。
4. The Definite Integral | 定积分
The definite integral ∫ₐᵇ f(x) dx is read as 'the integral from a to b of f(x) dx'. a is the lower limit and b the upper limit. Geometrically, it gives the net area — areas above the x-axis count positively, those below count negatively.
定积分 ∫ₐᵇ f(x) dx 读作“从 a 到 b，f(x) dx 的积分”。a 为下限，b 为上限。从几何意义上看，它给出的是净面积 — x 轴上方的面积取正，下方的取负。
ty (∫ (kf ± g) dx = k∫f dx ± ∫g dx), interval additivity (∫ₐᵇ f dx = ∫ₐᶜ f dx + ∫ᶜᵇ f dx), and reversal of limits (∫ₐᵇ f dx = −∫ᵇₐ f dx).
定积分的主要性质包括：线性性（∫ (kf ± g) dx = k∫f dx ± ∫g dx），区间可加性（∫ₐᵇ f dx = ∫ₐᶜ f dx + ∫ᶜᵇ f dx），以及交换限号变号（∫ₐᵇ f dx = −∫ᵇₐ f dx）。
5. The Fundamental Theorem of Calculus | 微积分基本定理
inks differentiation and integration as inverse processes. Part 1 states: if F(x) = ∫ₐˣ f(t) dt, then F' hich is our workhorse for computing areas, says that if F is any antiderivative of f, then
微积分基本定理（FTC）将微分与积分紧密联系为互逆的运算。第一部分指出：若 F(x) = ∫ₐˣ f(t) dt，则 F'(x) = f(x)。第二部分，也就是我们计算面积的主力工具，表明如果 F 是 f 的任一原函数，那么
∫ₐᵇ f(x) dx = F(b) − F(a)
This means that calculating a definite integral reduces to finding an antiderivative and evaluating it at the upper and lower limits.
这意味着计算定积分就归结为求出一个原函数，并在上限和下限处求值作差。
6. Evaluating Definite Integrals | 计算定积分
: find an antiderivative F(x) of f(x), then compute F(b) − F(a). For example, ∫₀¹ x² dx = [x³/3]₀¹ = (1/3) − 0 = 1/3.
标准步骤：求出 f(x) 的一个原函数 F(x)，然后计算 F(b) − F(a)。例如，∫₀¹ x² dx = [x³/3]₀¹ = (1/3) − 0 = 1/3。
With basic functions the antiderivatives are familiar: ∫ xⁿ dx = xⁿ⁺¹/(n+1) for n ≠ −1, ∫ eˣ dx = e&# egrals of polynomials, exponentials, and trigonometric functions are then straightforward.
基本函数的原函数是我们熟知的：∫ xⁿ dx = xⁿ⁺¹/(n+1)（n ≠ −1），∫ eˣ dx = eˣ，∫ sin x dx = −cos x 等。于是，多项式、指数函数和三角函数的定积分计算就变得直接了当。
7. Area Below the x-Axis | x 轴下方的面积
When f(x)
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图