Integration for Area Under a Curve | 曲线下面积的积分计算

📚 Integration for Area Under a Curve | 曲线下面积的积分计算

The area under a curve is one of the most fundamental concepts in calculus. In IB Mathematics, this idea is formalised through the definite integral, which links the geometric notion of area to the algebraic process of anti-differentiation. This article explains how to compute areas using integration, from the definition by Riemann sums to practical techniques for handling signed areas and regions between curves.

曲线下方的面积是微积分中最基本的概念之一。在IB数学中，这一思想通过定积分被正式化，将面积的几何概念与求原函数的代数过程联系起来。本文将解释如何使用积分计算面积，从黎曼和定义到处理有符号面积和曲线间区域的实际技巧。

1. The Definite Integral as Area | 定积分与面积

Given a continuous function \(f(x) \ge 0\) on an interval \([a, b]\), the area of the region bounded by the curve, the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\) is defined as the definite integral:

给定区间 \([a, b]\) 上的连续函数 \(f(x) \ge 0\)，由曲线、\(x\) 轴以及直线 \(x = a\) 和 \(x = b\) 所围成的区域面积定义为定积分：

Area = ∫ₐᵇ f(x) dx

The symbol ∫ represents the integral sign, \(a\) and \(b\) are the lower and upper limits, and \(dx\) indicates that we integrate with respect to \(x\).

符号 ∫ 表示积分号，\(a\) 和 \(b\) 分别是下限和上限，\(dx\) 表示我们关于 \(x\) 进行积分。

2. Riemann Sums and Approximations | 黎曼和与近似

The definite integral is originally defined as the limit of Riemann sums. We divide \([a, b]\) into \(n\) equal subintervals of width \(\Delta x = \frac{b-a}{n}\). On each subinterval, we choose a sample point \(x_i^*\) and form the sum:

定积分最初定义为黎曼和的极限。我们将 \([a, b]\) 分成 \(n\) 个等宽子区间，宽度 \(\Delta x = \frac{b-a}{n}\)。在每个子区间上选取一个样本点 \(x_i^*\)，构造和式：

∫ₐᵇ f(x) dx = limₙ→∞ ∑ᵢ₌₁ⁿ f(x_i^*) Δx

Using left endpoints, right endpoints, or midpoints gives different approximations. As \(n\) increases, these approximations converge to the exact area. In IB, you may be asked to estimate area using trapeziums or Simpson’s rule.

使用左端点、右端点或中点会得到不同的近似值。当 \(n\) 增大时，这些近似值收敛到精确面积。在IB中，你可能会被要求使用梯形法或辛普森法则来估计面积。

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

The Fundamental Theorem of Calculus provides a powerful method to evaluate definite integrals without using limits. It states that if \(F\) is an antiderivative of \(f\), then:

微积分基本定理提供了一种不用求极限就能计算定积分的强大方法。它指出若 \(F\) 是 \(f\) 的一个原函数，则：

∫ₐᵇ f(x) dx = F(b) − F(a)

Here \(F'(x) = f(x)\). This means we can find the exact area by evaluating the antiderivative at the upper limit and subtracting its value at the lower limit.

这里 \(F'(x) = f(x)\)。这意味着我们可以通过计算原函数在上限的值减去在下限的值来得到精确面积。

4. Indefinite Integrals and Basic Rules | 不定积分与基本规则

Before computing definite integrals, we need to master indefinite integrals. The power rule states:

在计算定积分之前，我们需要掌握不定积分。幂法则表述为：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

Other common rules include the constant multiple rule \(\int c f(x) dx = c \int f(x) dx\), the sum rule \(\int [f(x)+g(x)] dx = \int f(x) dx + \int g(x) dx\), and the special case \(\int \frac{1}{x} dx = \ln|x| + C\).

其他常见法则包括常数倍法则 \(\int c f(x) dx = c \int f(x) dx\)，和法则 \(\int [f(x)+g(x)] dx = \int f(x) dx + \int g(x) dx\)，以及特殊情况 \(\int \frac{1}{x} dx = \ln|x| + C\)。

5. Evaluating Definite Integrals | 计算定积分

To evaluate a definite integral, first find an antiderivative, then use the notation \([F(x)]ₐᵇ = F(b) − F(a)\). For example:

要计算定积分，首先找到原函数，然后使用记号 \([F(x)]ₐᵇ = F(b) − F(a)\)。例如：

∫₁³ (2x + 1) dx = [x² + x]₁³ = (9 + 3) − (1 + 1) = 10

Notice that the constant of integration \(C\) cancels out when subtracting, so it is omitted when evaluating definite integrals.

注意积分常数 \(C\) 在相减时抵消，因此在计算定积分时可以省略。

6. Signed Area vs Geometric Area | 有符号面积与几何面积

When \(f(x)\) is negative on part of the interval, the definite integral gives the signed area: positive above the \(x\)-axis, negative below. For example, \(\int₀^{2π} \sin x \, dx = 0\), because the positive and negative parts cancel.

当 \(f(x)\) 在区间的一部分为负时，定积分给出的是有符号面积：在 \(x\) 轴上方为正，下方为负。例如 \(\int₀^{2π} \sin x \, dx = 0\)，因为正负部分相互抵消。

To find the total geometric area, we must split the interval at the roots of \(f\) and take the absolute value of each integral:

要求总几何面积，我们必须将区间在 \(f\) 的零点处分开，并对每个积分取绝对值：

Total Area = ∫ₐᶜ f(x) dx − ∫ᶜᵇ f(x) dx

where \(c\) is a root of \(f\) in \((a, b)\).

其中 \(c\) 是 \(f\) 在 \((a, b)\) 内的一个零点。

7. Area Between Two Curves | 两条曲线之间的面积

If \(f(x) \ge g(x)\) on \([a, b]\), the area between the curves \(y = f(x)\) and \(y = g(x)\) is:

如果在 \([a, b]\) 上有 \(f(x) \ge g(x)\)，则曲线 \(y = f(x)\) 和 \(y = g(x)\) 之间的面积为：

Area = ∫ₐᵇ [f(x) − g(x)] dx

This formula works because the vertical distance between the curves is \(f(x) − g(x)\). Remember to identify which curve is on top over each subinterval. If the curves cross, split the integral at the intersection points.

这个公式成立是因为曲线之间的垂直距离是 \(f(x) − g(x)\)。记得要判断在每个子区间上哪条曲线在上方。如果曲线相交，需在交点处拆分积分。

8. Finding Intersection Points | 求交点

To apply the area-between-curves formula, we often need the x-coordinates of the intersection points. Solve the equation \(f(x) = g(x)\). For example, to find the area enclosed by \(y = x²\) and \(y = x + 2\):

要应用曲线间面积公式，我们常常需要交点的 x 坐标。解方程 \(f(x) = g(x)\)。例如，求 \(y = x²\) 和 \(y = x + 2\) 所围成的面积：

x² = x + 2 ⇒ x² − x − 2 = 0 ⇒ x = −1, 2

On \([-1, 2]\), the line \(y = x + 2\) lies above the parabola, so the area is \(\int_{−1}^{2} [(x + 2) − x²] dx\).

在 \([-1, 2]\) 上，直线 \(y = x + 2\) 位于抛物线之上，所以面积为 \(\int_{−1}^{2} [(x + 2) − x²] dx\)。

9. Integrating with Respect to y | 关于 y 积分

Sometimes it is easier to express curves as \(x = h(y)\). The area between two curves in the y-direction is:

有时将曲线表示为 \(x = h(y)\) 更容易。在 y 方向上两条曲线之间的面积为：

Area = ∫ᶜᵈ [x_right − x_left] dy

This is useful when the region is naturally bounded by horizontal lines and the curves are better described as functions of \(y\). For instance, the area between \(x = y²\) and \(x = y + 2\) uses limits \(y = −1\) and \(y = 2\).

当区域自然由水平线界定，且曲线更适合描述为 \(y\) 的函数时，这种方法很有用。例如，\(x = y²\) 与 \(x = y + 2\) 之间的面积使用下限 \(y = −1\) 和上限 \(y = 2\)。

10. Numerical Integration | 数值积分

When an antiderivative is difficult or impossible to find, we use numerical methods. The trapezium rule approximates the integral by dividing the area into trapeziums:

当原函数难以或无法找到时，我们使用数值方法。梯形法则通过将面积分成梯形来近似积分：

∫ₐᵇ f(x) dx ≈ (Δx/2) [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ)]

where \(Δx = \frac{b-a}{n}\) and \(x_i = a + iΔx\). The approximation becomes more accurate as \(n\) increases. In IB, you may also encounter Simpson’s rule, which uses quadratic approximations on pairs of intervals.

其中 \(Δx = \frac{b-a}{n}\)，且 \(x_i = a + iΔx\)。随着 \(n\) 增大，近似变得更加精确。在IB中，你还可能遇到辛普森法则，它在成对的区间上使用二次函数近似。

11. Common Pitfalls and IB Tips | 常见错误与IB提示

One common mistake is forgetting to split the integral when the curve crosses the x-axis. Another is using the wrong formula when the curves are not explicitly labelled as “top” and “bottom”. Always sketch the graph when possible.

一个常见错误是当曲线穿过 x 轴时忘记拆分积分。另一个是在曲线未明确标出”上方”和”下方”时使用了错误的公式。只要可能，务必画出图形。

Always check the sign of \(f(x)\) on the interval.

始终检查 \(f(x)\) 在区间上的符号。
For area between curves, subtract the lower function from the upper function.

对于曲线间的面积，用上方函数减去下方函数。
Use the symmetry of even or odd functions to simplify integrals.

利用偶函数或奇函数的对称性来简化积分。
Be careful with absolute values: \(\int |f(x)| dx\) requires splitting at zeros.

注意绝对值：\(\int |f(x)| dx\) 需要在零点处拆分。

12. Worked Example: Area Under a Cubic | 例题：三次曲线下的面积

Find the total area bounded by \(y = x³ − 4x\), the x-axis, and the lines \(x = −2\) and \(x = 3\).

求曲线 \(y = x³ − 4x\)、\(x\) 轴以及直线 \(x = −2\) 和 \(x = 3\) 所围成的总几何面积。

First factor: \(x³ − 4x = x(x − 2)(x + 2)\). Roots are at \(x = −2, 0, 2\). On \([-2, 0]\), the function is positive? Test \(x = −1\): \(y = 3\) (positive). On \([0, 2]\), test \(x = 1\): \(y = −3\) (negative). On \([2, 3]\), test \(x = 2.5\): \(y = 5.625\) (positive). Therefore:

首先因式分解：\(x³ − 4x = x(x − 2)(x + 2)\)。零点在 \(x = −2, 0, 2\)。在 \([-2, 0]\) 上，函数为正？取 \(x = −1\)：\(y = 3\)（正）。在 \([0, 2]\) 上，取 \(x = 1\)：\(y = −3\)（负）。在 \([2, 3]\) 上，取 \(x = 2.5\)：\(y = 5.625\)（正）。因此：

Total Area = ∫_{−2}^{0} (x³ − 4x) dx − ∫₀² (x³ − 4x) dx + ∫₂³ (x³ − 4x) dx

Compute each integral: \(\left[\frac{x⁴}{4} − 2x²\right]\) evaluated at the limits gives respectively \(\frac{16}{4}−8 = −4\), so first part is \(+4\); the second part gives \(-4\), so minus that gives \(+4\); the third part: \(\frac{81}{4}−18 − \left(\frac{16}{4}−8\right) = 2.25 + 4 = 6.25\). Total area \(= 4 + 4 + 6.25 = 14.25\).

分别计算每个积分：\(\left[\frac{x⁴}{4} − 2x²\right]\) 在对应端点处的值分别为 \(\frac{16}{4}−8 = −4\)，所以第一部分为 \(+4\)；第二部分得到 \(-4\)，减去它得到 \(+4\)；第三部分：\(\frac{81}{4}−18 − \left(\frac{16}{4}−8\right) = 2.25 + 4 = 6.25\)。总面积为 \(4 + 4 + 6.25 = 14.25\)。

Mastering integration for area calculations requires practice with algebraic antiderivatives, careful analysis of intervals, and a solid understanding of signed vs. geometric area. Keep these principles in mind, and you will solve IB area problems confidently.

掌握用于面积计算的积分需要练习代数原函数、仔细分析区间，并深刻理解有符号面积与几何面积的区别。牢记这些原则，你就能自信地解决IB面积问题。

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Integration for Area Under a Curve | 曲线下面积的积分计算