Work through the list without notes. Use a calculator only where the linked question allows one. 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 height of a strip is |f(x) − g(x)| when the curves cross. (Guide: “When the top curve changes”, Figure 1)
- I can explain the difference between ∫ (f − g) dx and ∫ |f − g| dx, and say which one is the area. (Guide: “Net value or area?”; Practice Q1, Q5(d))
- I can explain why |∫ (f − g) dx| is not the area when the curves cross. (Guide: “Common misconceptions”; Practice Q1, Q3)
Skills
- I can find every crossing of two curves by factoring, including a crossing at x = 0. (Guide: Worked example 1; Practice Q2, Q5(a))
- I can make a sign chart for f − g with one test value per piece. (Guide: “The method, step by step”; Practice Q5(b), Q6(b))
- I can write an area as a sum of integrals, using top − bottom on each piece, and evaluate it exactly. (Guide: Worked example 1; Practice Q2, Q5(c))
- I can find crossings with a calculator and evaluate ∫ |f(x) − g(x)| dx. (Guide: Worked example 2; Practice Q4, Q6(c))
- I can split a dy integral where the right and left curves swap. (Practice Q7)
Reasoning
- I can use given integral values to decide which curve is on top and to find a total area or a net value. (Guide: Worked example 3; Practice Q3)
- I can explain why symmetric pieces can make a net integral equal to 0 even though the area is not 0. (Practice Q4, Q5(d))
- I can use an area in a real context, such as finding a mass from an area. (Practice Q6(d))
All ticked? Move on to Topic 8.7, Volumes with Cross Sections: Squares and Rectangles, or return to the Calculus AB roadmap or the Calculus BC roadmap.