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 of a solid is the integral of its cross-sectional area, using thin slabs of volume A(x) Δx. (Guide: “The idea: a solid is a stack of thin slabs”)
- I can check the formula on a prism: equal slices give area × length. (Guide: “A quick check”; Practice Q7(c))
- I can describe where a cross section sits: standing at right angles to the base, on a segment across it. (Guide: Figures 1 and 2)
Calculation
- I can find the side length s as top − bottom for slices perpendicular to the x-axis. (Guide: Worked example 1; Practice Q1, Q2, Q5(a))
- I can find the side length s as right − left, in terms of y, for slices perpendicular to the y-axis, and integrate with dy. (Guide: Worked example 2; Practice Q3)
- I can write A = s² for squares and A = s × height for rectangles, including heights given in words. (Guide: area table; Practice Q4, Q6)
- I can evaluate a volume integral exactly, expanding (top − bottom)² first. (Guide: Worked example 1; Practice Q5(b))
- I can find a volume with a calculator, storing intersection values and giving three decimal places. (Guide: Worked example 3; Practice Q4, Q6)
Reasoning
- I can explain why (f − g)² and f² − g² give different volumes, and which one is right here. (Guide: “Common misconceptions”; Practice Q2)
- I can show that rectangles of constant height h give a volume of h × (area of the base). (Practice Q5(c))
- I can interpret a cross-sectional area and a volume in context, with square and cubic units. (Practice Q6)
- I can find an unknown constant from a given volume. (Practice Q7(b))
All ticked? Move on to Volumes with Cross Sections: Triangles and Semicircles, or return to the Calculus AB roadmap or the Calculus BC roadmap.