📚 Line Integrals along Paths Parallel to Coordinate Axes | IB数学：平行于坐标轴的线积分路径
A line integral along an axis-parallel path is one of the simplest ways to learn about integration over curves. When the path is horizontal or vertical, the general expression ∫C Pdx + Qdy reduces immediately to an ordinary one-variable integral.
沿着平行于坐标轴的路径计算线积分，是学习曲线积分最直观的入门方式。当路径为水平或竖直时，一般表达式 ∫C Pdx + Qdy 会立刻化简为普通的一元积分。
1. What is a Line Integral? | 什么是线积分？
For a vector field F = P(x,y)i + Q(x,y)j, the line integral of F along an oriented curve C is defined as
对于向量场 F = P(x,y)i + Q(x,y)j，沿有向曲线 C 的线积分定义为
∫C F · dr = ∫C P dx + Q dy
Here dr = (dx, dy), and the integral is often called the circulation or work integral, depending on the physical context.
其中 dr = (dx, dy)。这个积分在物理中常被称为环量或做功积分。
If the curve is made of segments parallel to the coordinate axes, then on every segment either dx = 0 or dy = 0. This removes one term and makes the integration straightforward.
如果曲线由平行于坐标轴的线段组成，那么在每一段上要么 dx = 0，要么 dy = 0。这会使积分中的一项消失，从而使计算变得非常直接。
2. Paths Parallel to the Axes | 平行于坐标轴的路径
A horizontal path is described by y = c, where c is a constant. Along such a path, dy = 0.
水平路径可以表示为 y = c，其中 c 是常数。在这条路径上，dy = 0。
A vertical path is described by x = c, where c is a constant. Along such a path, dx = 0.
竖直路径可以表示为 x = c，其中 c 是常数。在这条路径上，dx = 0。
These two cases are the building blocks for more complicated piecewise paths.
这两种情况是构造更复杂分段路径的基本单元。
| Path type | Condition | Simplified integral |
| Horizontal | y = c, dy = 0 | ∫ab P(x,c)dx |
| Vertical | x = c, dx = 0 | ∫y1y2 Q(c,y)dy |
Notice that on a horizontal segment only the P component contributes; on a vertical segment only the Q component contributes.
注意：在水平线段上只有 P 分量有贡献；在竖直线段上只有 Q 分量有贡献。
3. Horizontal Path: y = Constant | 水平路径：y 为常数
Suppose C is a horizontal segment from (a,c) to (b,c). Since y = c, we have dy = 0.
设 C 是从 (a,c) 到 (b,c) 的水平线段。因为 y = c，所以 dy = 0。
∫C F · dr = ∫ab P(x,c) dx
For example, let F = (xy, x2) and take the path from (1,2) to (4,2). Here y = 2, so
例如，设 F = (xy, x2)，路径是从 (1,2) 到 (4,2)。此时 y = 2，于是
∫14 x(2) dx = ∫14 2x dx = x²
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