Polar curves and area
| English | Português |
|---|---|
| polar coordinate/ˈpəʊlə kəʊˈɔːdɪnət/ | coordenada polar |
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?
A position described by distance r and angle θ from a chosen origin and reference ray.
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
Use x=r cosθ and y=r sinθ
Compare the model with the worked case and explain one change.
For r=2,θ=π/3, find x.
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.
Area/π=r²=4 for the circle.
Using any full parameter interval always counts each polar region exactly once.
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.
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.
A half-turn encloses half the circle: area/π=2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further 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.