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 how thin vertical strips of height top − bottom and width Δx lead to the area integral. (Guide: “From area under a curve to area between curves”, Figure 1)
- I can explain why top − bottom gives the area even when the region lies partly or wholly below the x-axis. (Guide: “Why the x-axis does not matter”; Practice Q5(d))
- I can explain why a negative result means the subtraction was the wrong way round. (Guide: Worked example 1 check; Practice Q2)
- I can explain when an area must be split into two integrals. (Guide: Worked example 3)
Skills
- I can find the limits of integration by solving f(x) = g(x) algebraically. (Guide: Worked example 1; Practice Q1, Q5(a))
- I can find intersection points with a calculator, store them and use them as limits. (Guide: Worked example 2; Practice Q4, Q6(a))
- I can decide which curve is on top using a test value between the limits. (Guide: “The method, step by step”; Practice Q5(b))
- I can write an area integral with correct limits, brackets and dx before evaluating it. (Guide: Worked example 2 notation; Practice Q1, Q6(b))
- I can evaluate an area exactly, including with trigonometric functions. (Guide: Worked example 1; Practice Q2, Q3)
Reasoning
- I can find the area between a curve and the x-axis, treating the axis as y = 0. (Guide: “A curve and the x-axis”; Revision notes self-check 2)
- I can interpret the height of a strip, f(x) − g(x), in a real context. (Practice Q6(d))
- I can set up an area that depends on a constant and solve for the constant. (Practice Q7)
All ticked? Move on to Topic 8.5, Finding the Area Between Curves Expressed as Functions of y, or return to the Calculus AB roadmap or the Calculus BC roadmap.