Polar curves and area · 极坐标曲线与面积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| polar coordinate/ˈpəʊlə kəʊˈɔːdɪnət/ | 极坐标 | jí zuò biāo |
Why describe a curve by direction?
- A petal-shaped curve is simpler when its radius depends on direction. Cartesian coordinates can hide this structure.
- This lesson studies polar coordinate 极坐标: A position described by distance r and angle θ from a chosen origin and reference ray.
Choose the mathematical structure
- Use x=r cosθ and y=r sinθ. A polar area is one half the integral of r² with respect to θ over a correctly chosen interval. Identify symmetry and repeated tracing before selecting limits.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines polar coordinate? · 哪种描述正确定义了极坐标？
A position described by distance r and angle θ from a chosen origin and reference ray. · 由选定原点和参考射线确定的距离 r 和角度 θ 描述的位置。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For r=2 and θ=π/3, x=1 and y=√3. For the circle r=2 over a complete turn, A=(1/2) integral_0^(2π) 4 dθ=4π. A half-turn gives 2π, exactly half the disk.
Polar curves and area · 极坐标曲线与面积
Use x=r cosθ and y=r sinθ · 使用 x=r cosθ 和 y=r sinθ
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
For r=2,θ=π/3, find x. · 当 r=2, θ=π/3 时，求 x。
x=r cosθ=2 cos(π/3)=1. · x=r cosθ=2 cos(π/3)=1。
Test a tempting shortcut
- Negative r places a point in the opposite direction; it is not an ordinary negative distance along the same ray. A parametrisation may trace the same region more than once, so a full parameter interval can overcount area.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Using any full parameter interval always counts each polar region exactly once. This claim is false. Explain which definition or assumption it violates.
For a full circle r=2, find area divided by π. · 对于整圆 r=2，求面积除以 π 的值。
Area/π=r²=4 for the circle. · 圆的面积/π = r² = 4。
Using any full parameter interval always counts each polar region exactly once. · 使用任意完整的参数区间总是能恰好计算每个极坐标区域一次。
Negative r places a point in the opposite direction; it is not an ordinary negative distance along the same ray. A parametrisation may trace the same region more than once, so a full parameter interval can overcount area. · 负的 r 将点置于相反方向；它不是沿同一条射线的普通负距离。参数化可能多次描绘同一区域，因此完整的参数区间可能导致面积重复计数。
Interpret a new situation
- Sketch enough points to establish orientation and bounds. For an enclosed region between two curves, determine intersections and which radial square contributes the outer boundary on each interval.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For that circle over a half-turn, find area divided by π. · 对于该圆超过半周的部分，求面积除以 π。
A half-turn encloses half the circle: area/π=2. · 半周包围了半个圆：面积/π=2。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A position described by distance r and angle θ from a chosen origin and reference ray. Choose the relationship, show the method, check its assumptions and interpret the result.