📚 3 Repeated integrals over non-rectangular regions | 3 非矩形区域上的累次积分
When we first learn double integrals, they are usually presented over rectangular domains, where the limits of integration are constants. This is the simplest case, but real-world regions – such as triangles, circles, and irregular shapes – are rarely rectangles. To handle these, we must extend the concept of repeated (iterated) integrals to non‑rectangular regions, carefully describing variable limits that depend on the order of integration. Mastery of this technique is essential for applications in probability, physics, and engineering, and it forms a core part of the IB HL calculus syllabus.
当我们最初学习二重积分时，通常是在矩形区域上进行，积分上下限都是常数。这是最简单的情形，但现实世界中的区域——例如三角形、圆形和不规则形状——很少是矩形。为了处理这些区域，我们必须将累次（迭代）积分的概念扩展到非矩形区域，并根据积分顺序准确描述变量上下限。掌握这项技术对于概率、物理和工程中的应用至关重要，也是 IB 高等数学微积分部分的核心内容。
1. From Rectangular to General Regions | 从矩形区域到一般区域
For a rectangular domain R = [a, b] × [c, d], the double integral ∬R f(x, y) dA is simply computed as ∫x=ab ∫y=cd f(x, y) dy dx. The limits are constant and order does not fundamentally trouble us. However, when the region D is bounded by curves, one or both inner limits become functions of the outer variable. The key is to view a non‑rectangular region as either vertically simple (Type I) or horizontally simple (Type II), and then set up iterated integrals accordingly.
对于矩形区域 R = [a, b] × [c, d]，二重积分 ∬R f(x, y) dA 可以直接计算为 ∫x=ab ∫y=cd f(x, y) dy dx。此时积分限是常数，积分次序不会带来根本性困难。然而，当区域 D 由曲线围成时，一个或两个内层积分限将成为外层变量的函数。关键在于将非矩形区域视为垂直简单（类型 I）或水平简单（类型 II），然后相应地建立累次积分。
2. Type I Regions: Vertically Simple | 类型 I 区域：垂直简单
A plane region D is called Type I if it can be described as D = {(x, y) | a ≤ x ≤ b, g₁(x) ≤ y ≤ g₂(x)}, where g₁ and g₂ are continuous functions on [a, b]. For each fixed x between a and b, y runs from the lower boundary curve y = g₁(x) to the upper boundary y = g₂(x). The corresponding double integral is then expressed as an iterated integral in the order dy dx:
若平面区域 D 可以表示为 D = {(x, y) | a ≤ x ≤ b, g₁(x) ≤ y ≤ g₂(x)}，其中 g₁ 和 g₂ 在 [a, b] 上连续，则称 D 为类型 I。对于 a 与 b 之间每一个固定的 x，y 从下边界曲线 y = g₁(x) 到上边界 y = g₂(x) 变化。相应的二重积分可表示为 dy dx 顺序的累次积分：
∬D f(x, y) dA = ∫x=ab ∫y=g₁(x)g₂(x) f(x, y) dy dx
Always draw a diagram and perform the “vertical line test”: a vertical line through the region should enter at y = g₁(x) and exit at y = g₂(x). If this holds for every x in [a, b], the region is Type I. Examples include triangles with a slanted side, regions bounded by a parabola and a line, or any shape where top and bottom are functions of x.
务必画出简图并进行“垂直线测试”：穿过区域的垂直线应在 y = g₁(x) 处进入，在 y = g₂(x) 处离开。如果对 [a, b] 中的每个 x 都成立，该区域即为类型 I。例如带有斜边的三角形、由抛物线和直线围成的区域，或任何上下边界均为 x 的函数的形状。
3. Type II Regions: Horizontally Simple | 类型 II 区域：水平简单
A region D is Type II if it can be written as D = {(x, y) | c ≤ y ≤ d, h₁(y) ≤ x ≤ h₂(y)}, where h₁ and h₂ are continuous functions of y. Here we fix y and let x vary between the left and right boundaries. The double integral then takes the order dx dy:
若区域 D 可表示为 D = {(x, y) | c ≤ y ≤ d, h₁(y) ≤ x ≤ h₂(y)}，其中 h₁ 与 h₂ 是 y 的连续函数，则称为类型 II。此时固定 y，让 x 在左、右边界之间变化。二重积分以 dx dy 顺序写出：
∬D f(x, y) dA = ∫y=cd ∫x=h₁(y)h₂(y) f(x, y) dx dy
A horizontal line test determines if a region is Type II: any horizontal line intersecting the region should enter at x = h₁(y) and exit at x = h₂(y). Many regions are both Type I and Type II, giving us the freedom to choose the more convenient order of integration.
通过水平线测试可判断一个区域是否为类型 II：任何与区域相交的水平线应在 x = h₁(y) 处进入，在 x = h₂(y) 处离开。许多区域同时是类型 I 和类型 II，这让我们可以选择更方便的积分次序。
4. Setting Up Limits from a Sketch | 根据草图确定积分限
Begin by sketching the bounding curves and finding their intersection points. Decide whether to treat the region as Type I or Type II. For Type I, identify the left‑most and right‑most x‑values (a and b). Then draw a vertical line at a generic x; read the bottom curve as y = g₁(x) and the top as y = g₂(x). For Type II, find the lowest and highest y‑values (c and d), and express the left and right boundaries as functions of y. A common pitfall is using the wrong variable in the limits – always ensure the inner limits involve only the outer variable.
首先画出边界曲线的草图并求出它们的交点。决定将区域视为类型 I 还是类型 II。对于类型 I，确定最左和最右的 x 值（a 和 b）。然后在任意 x 处作竖直线；读出下边界曲线 y = g₁(x) 和上边界 y = g₂(x)。对于类型 II，找出最低和最高的 y 值（c 和 d），并将左右边界表示为 y 的函数。常见错误是在积分限中使用错误变量——应始终确保内层积分限只依赖于外层变量。
5. Changing the Order of Integration | 交换积分次序
Sometimes an integral like ∫01 ∫y1 ex² dx dy is impossible to evaluate directly because the inner integral has no elementary antiderivative. By reversing the order of integration, the problem becomes tractable. To reverse, sketch the region described by the original limits. For the example, the original limits say: 0 ≤ y ≤ 1 and y ≤ x ≤ 1, which describes a triangular region. Re‑describing it as Type I gives 0 ≤ x ≤ 1, 0 ≤ y ≤ x, so the integral becomes ∫01 ∫0x ex² dy dx, easily evaluated by first integrating with respect to y. Always re‑sketch and re‑label the bounds carefully.
有时像 ∫01 ∫y1 ex² dx dy 这样的积分无法直接计算，因为内层积分没有初等原函数。通过交换积分次序，问题便可解决。要交换次序，首先画出原积分限所描述的区域。在上述例子中，原限表示 0 ≤ y ≤ 1 且 y ≤ x ≤ 1，描述了一个三角形区域。将其重新描述为类型 I 得到 0 ≤ x ≤ 1，0 ≤ y ≤ x，于是积分变为 ∫01 ∫0x ex² dy dx，先对 y 积分便能轻松求出。每次交换次序都要重新画图并仔细标记边界。
6. Polar Coordinates for Circular Symmetry | 利用极坐标处理圆对称性
For regions with circular or radial symmetry – disks, sectors, cardioids – polar coordinates (r, θ) often simplify the integrals dramatically. Recall that x = r cos θ, y = r sin θ, and the area element transforms as dA = r dr dθ. A typical type in IB is integrating over a circle or an annulus. The double integral takes the form:
对于具有圆对称性或辐射对称性的区域——如圆盘、扇形、心形线——极坐标 (r, θ) 往往能极大简化积分。回忆 x = r cos θ, y = r sin θ，面积微元变换为 dA = r dr dθ。IB 中常见的是在圆或圆环上积分。二重积分的形式为：
∬D f(x, y) dA = ∫θ=αβ ∫r=r₁(θ)r₂(θ) f(r cos θ, r sin θ) r dr dθ
Here the radial limits r₁(θ) and r₂(θ) are often constants for a full disk or functions of θ for more exotic shapes. Do not forget the extra factor r – it is one of the most frequent mistakes.
这里的径向限 r₁(θ) 和 r₂(θ) 对于整个圆盘通常是常数，对于更奇特的形状则为 θ 的函数。切勿漏掉额外的 r 因子——这是最常犯的错误之一。
7. Computing Areas and Volumes | 计算面积与体积
If f(x, y) = 1, the double integral over D gives the area of D: A = ∬D 1 dA. This can be evaluated either as a Type I or Type II iterated integral. For example, the area of a region bounded by y = x² and y = 2x + 3 is found by setting up limits from the intersection points. When f(x, y) ≥ 0 represents a surface height, the double integral computes the volume under the surface z = f(x, y) above region D. Repeated integrals thus unify area and volume calculations under one framework.
若 f(x, y) = 1，则区域 D 上的二重积分给出 D 的面积：A = ∬D 1 dA。这可以按类型 I 或类型 II 的累次积分来求值。例如，由 y = x² 和 y = 2x + 3 围成的区域，其面积可通过从交点出发设定积分限来求得。当 f(x, y) ≥ 0 表示曲面的高度时，二重积分便计算了曲面 z = f(x, y) 在区域 D 上方的体积。因此累次积分将面积与体积的计算统一在同一个框架之下。
8. Worked Example: Triangular Region | 示例：三角形区域
Evaluate ∬D xy dA, where D is the triangle with vertices (0,0), (1,0) and (0,2). This region is both Type I and Type II. Using the Type I description: x runs from 0 to 1. For a fixed x, y runs from the lower boundary y = 0 to the line joining (0,2) and (1,0), which has equation y = 2 – 2x. Hence the integral is
计算 ∬D xy dA，其中 D 是以 (0,0), (1,0) 和 (0,2) 为顶点的三角形。该区域同为类型 I 和类型 II。采用类型 I 描述：x 从 0 到 1。对于固定的 x，y 从下边界 y = 0 到连接 (0,2) 与 (1,0) 的直线，其方程为 y = 2 – 2x。因此积分为
∬D xy dA = ∫01 ∫02-2x xy dy dx.
First integrate with respect to y: ∫ xy dy = ½ x y², evaluated from 0 to 2-2x gives ½ x (2-2x)² = 2x(1-x)². Then integrate with respect to x from 0 to 1: ∫01 2x(1 – 2x + x²) dx = 2 ∫01 (x – 2x² + x³) dx = 2 [½ – 2/3 + 1/4] = 2 [6/12 – 8/12 + 3/12] = 2 (1/12) = 1/6. The result is 1/6.
先对 y 积分：∫ xy dy = ½ x y²，从 0 到 2-2x 得 ½ x (2-2x)² = 2x(1-x)²。然后对 x 从 0 到 1 积分：∫01 2x(1 – 2x + x²) dx = 2 ∫01 (x – 2x² + x³) dx = 2 [½ – 2/3 + 1/4] = 2 [6/12 – 8/12 + 3/12] = 2 (1/12) = 1/6。结果为 1/6。
9. Worked Example: Region Bounded by a Parabola and a Line | 示例：抛物线与直线围成的区域
Consider ∬D (x + y) dA, where D is the region enclosed by y = x² and y = x + 2. Find intersection points: x² = x + 2 → x² – x – 2 = 0 → x = -1, 2. For a Type I region, x runs from -1 to 2, and y goes from the lower curve y = x² to the upper line y = x + 2. The iterated integral is ∫-12 ∫y=x²x+2 (x + y) dy dx. Integrate dy first: ∫ (x + y) dy = xy + ½ y², substitute limits to obtain (x(x+2) + ½ (x+2)²) – (x·x² + ½ (x²)²). Simplify and integrate with respect to x. The computation yields a numerical answer that can be verified by reversing the order, giving the same result. Always check that the region is correctly captured by the chosen limits.
考虑 ∬D (x + y) dA，其中 D 是由 y = x² 和 y = x + 2 所围成的区域。求交点：x² = x + 2 → x² – x – 2 = 0 → x = -1, 2。对于类型 I 区域，x 从 -1 到 2，y 从下曲线 y = x² 到上直线 y = x + 2。累次积分为 ∫-12 ∫y=x²x+2 (x + y) dy dx。先对 y 积分：∫ (x + y) dy = xy + ½ y²，代入积分限得 (x(x+2) + ½ (x+2)²) – (x·x² + ½ (x²)²)。化简后再对 x 积分。计算可得出数值结果，并可通过交换次序加以验证，两者结果应一致。务必检查所选的积分限是否正确描述了该区域。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One of the most frequent errors is forgetting that the inner limits must be functions of the outer variable, not the inner one. When writing a Type I integral, the y‑limits must be expressed in terms of x; writing them in terms of y leads to nonsense. Another error is misidentifying which curve is upper/lower or right/left – always test with a point inside the region. In polar coordinates, omitting the extra r or using dx dy instead of r dr dθ will produce a dimensionally wrong answer. Finally, when reversing the order, do not simply swap the integrals; you must redraw the region and re‑express the limits accordingly.
最常犯的一个错误是忘记内层积分限必须是外层变量的函数，而不能是内层变量的函数。在写类型 I 积分时，y 的上下限必须用 x 表示；若用 y 来表示就会产生错误。另一个错误是混淆哪条曲线是上/下边界或左/右边界——始终用区域内一点检验。在极坐标中，漏掉额外的 r 或用 dx dy 代替 r dr dθ 会导致量纲错误。最后，交换次序时不可简单地对调积分符号；必须重新画图并据此重新表达积分限。
11. Summary and Further Connections | 总结与延伸联系
Repeated integrals over non‑rectangular regions are a gateway to multivariable calculus. By mastering the art of describing a region in both Type I and Type II forms, you gain flexibility to choose the most efficient integration order. Polar coordinates add another layer of power for circular domains. These ideas directly feed into triple integrals, surface integrals, and the calculation of centres of mass and moments of inertia in physics. In IB exams, clarity of setting up limits and careful algebraic simplification are always rewarded. Remember: sketch, label, integrate step‑by‑step, and check.
非矩形区域上的累次积分是进入多变量微积分的大门。通过掌握将区域表述为类型 I 和类型 II 的技巧，你可以灵活选择最有效的积分次序。极坐标则为圆形区域增添了又一层威力。这些思想直接通向三重积分、曲面积分以及物理学中质心和转动惯量的计算。在 IB 考试中，清晰设定积分限和细致的代数化简总能赢得分数。请记住：画草图，标边界，逐步积分，并复查。
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