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 (chain rule, Mean Value Theorem 5.1, Riemann sums 6.2 to 6.6, FTC 6.7) are on the Calculus BC roadmap.
Understanding
- I can explain how adding the lengths of many short chords, using Pythagoras and the Mean Value Theorem, leads to the arc length integral. (Guide: “Building the formula from straight pieces”)
- I can state what “smooth” means for this formula and what to do at a corner. (Guide: “The arc length formula”)
- I can explain why the distance traveled along a path equals the arc length, and how it differs from the chord, the vertical change and the total distance of Topic 8.2. (Guide: “Arc length as distance traveled”; Practice Q2(D), Q6(c))
Skills
- I can write the arc length integral for a curve y = f(x), squaring the derivative and keeping the 1 and the square root. (Guide: “The arc length formula”; Practice Q1)
- I can find an exact length by hand when 1 + (f′)² simplifies, including a chain-rule factor. (Guide: Worked example 1; Practice Q2)
- I can show that 1 + (f′)² is a perfect square and use it to find an exact length. (Practice Q5)
- I can find the length of a curve x = g(y), using y-limits and dx/dy. (Guide: “Curves written as x = g(y)”; Practice Q3)
- I can set up an arc length integral by hand and evaluate it with a calculator to 3 decimal places. (Guide: Worked example 2; Practice Q6(b), Q7(c))
Reasoning
- I can check an arc length against b − a and the straight-line distance between the end points. (Guide: “Three quick facts”; Practice Q4)
- I can justify upper and lower bounds on a length from bounds on the integrand. (Guide: Worked example 2 checks; Practice Q6(d))
- I can use arc length to answer a distance-traveled question in context, with units, including time at a given speed. (Guide: Worked example 2; Practice Q6(c))
- I can write a distance-along-the-path function s(x) as an integral, find s′(x), and solve s(x) = k with a calculator. (Practice Q7)
All ticked? Move on to Topic 9.1, defining and differentiating parametric equations, or return to the Calculus BC roadmap.