Work through the list without notes. Use a calculator only where a statement says so. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain why spinning a region about an axis it touches gives circular slices, and link this to the cross-section method of Topics 8.7 and 8.8. (Guide: “From known cross sections to solids of revolution”; Figure 1)
- I can identify the radius of a disc as the distance from the axis to the curve, measured at right angles to the axis. (Guide: “The disc method”; Practice Q5(c))
- I can explain why a curve below the x-axis still gives a positive disc volume. (Guide: “A curve below the axis”)
- I can tell when the disc method does not apply because the region does not touch the axis. (Guide: “The disc method”, point 3; Practice Q7(c))
Skills
- I can write π ∫ (a to b) [f(x)]² dx for a rotation about the x-axis, with π, the square and x-limits. (Guide: Worked example 1; Practice Q1, Q2)
- I can rewrite a curve as x = g(y) and write π ∫ (c to d) [g(y)]² dy for a rotation about the y-axis, with y-limits. (Guide: Worked example 2; Practice Q3, Q5(b))
- I can expand a squared radius such as (3x − x²)² before integrating, and evaluate the volume exactly. (Guide: Worked example 1; Practice Q5(a))
- I can write the integral in full and then evaluate a disc volume with a calculator to three decimal places. (Guide: Worked example 3; Practice Q4, Q6(c))
- I can estimate a volume from a table of radii with a trapezoidal sum of the areas πr². (Practice Q6(b))
Reasoning
- I can check a volume against a cylinder built from the widest radius and the length of the solid. (Guide: checks in Worked examples 1 to 3)
- I can explain why π[∫ f(x) dx]², 2π ∫ [f(x)]² dx and π ∫ f(x) dx are not disc volumes. (Guide: “Common misconceptions”; Practice Q1, Q2)
- I can find an unknown limit from a given volume. (Practice Q7(a), Q7(b))
- I can interpret a volume of revolution in context with cubic units. (Practice Q6(d))
All ticked? Move on to Topic 8.10, Volume with the Disc Method: Revolving Around Other Axes, or return to the Calculus AB roadmap or the Calculus BC roadmap.