BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 9.1, 8.13, 6.9) are on the Calculus BC roadmap.
Understanding
- I can explain, with a sketch of a small right triangle, why the length of a short piece of curve is about √(x′(t)² + y′(t)²) Δt. (Guide: “The idea: Pythagoras on tiny pieces”)
- I can state the two conditions for the integral to give a curve’s length: smooth, and traced exactly once. (Guide: “When does the integral give the length?”; Practice Q4)
- I can show that the Topic 8.13 formula for y = f(x) is a special case of the parametric formula. (Guide: “Link to Topic 8.13”)
Skills
- I can write the arc length integral for a parametric curve, with the chain rule applied and the limits in t. (Guide: “A strategy for any arc length question”; Practice Q1)
- I can find an exact length when (dx/dt)² + (dy/dt)² is a perfect square. (Guide: Worked example 1; Practice Q5, Q6)
- I can find the length of a circular arc from its parametric equations. (Practice Q2)
- I can evaluate an arc length integral with a calculator in radian mode, to 3 decimal places. (Guide: Worked example 2; Practice Q3, Q7(b))
Reasoning
- I can explain why an integral over a curve traced twice is not the curve’s length. (Practice Q4)
- I can prove a curve is traced once by showing x(t) or y(t) is always increasing or always decreasing. (Practice Q7(a))
- I can check an arc length against the straight-line distance between the end points and explain why it must be longer. (Guide: Worked example 2; Practice Q3, Q7(c))
- I can find where a parametric curve crosses itself and the length of the loop it makes. (Practice Q6)
All ticked? Move on to the Topic 9.4 study guide, Defining and Differentiating Vector-Valued Functions, or return to the Calculus BC roadmap.