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 the volume is still ∫ A dx when the slices are triangles or semicircles. (Guide: “Same integral, new shapes”)
- I can derive the area of an equilateral triangle of side s, (√3/4)s², using Pythagoras. (Guide: “The area formulas, derived”; Practice Q2)
- I can explain why a semicircle on side s has area (π/8)s², with radius s/2. (Guide: Figure 1; Practice Q1)
- I can tell the two isosceles right-triangle cases apart: leg in the base gives s²/2, hypotenuse gives s²/4. (Guide: area table; Practice Q3)
Calculation
- I can find a volume with triangular or semicircular slices perpendicular to the x-axis. (Guide: Worked example 1; Practice Q1, Q5)
- I can find a volume with slices perpendicular to the y-axis, writing s in terms of y. (Guide: Worked example 2; Practice Q2, Q7(c))
- I can use a calculator for the limits and the integral, storing values and giving three decimal places. (Guide: Worked example 3; Practice Q4, Q6(c))
- I can find a volume from a given area function, or from a triangle with a stated height. (Guide: “Other shapes and given areas”; Practice Q7(a))
- I can estimate a volume from a table with a trapezoidal sum of cross-sectional areas. (Practice Q6(b))
Reasoning
- I can explain why the ratio of two volumes on the same base depends only on the shape constants. (Practice Q5(d))
- I can find an unknown constant from a given volume. (Practice Q7(b))
- I can explain how these solids differ from solids of revolution in Topic 8.9. (Guide: “Common misconceptions”)
All ticked? Move on to Topic 8.9, Volume with Disc Method: Revolving Around the x- or y-Axis, or return to the Calculus AB roadmap or the Calculus BC roadmap.