Work through the list without notes or a calculator. 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 a definite integral is a signed area, and why it can be 0 or negative. (Guide: “Integrals from geometry”; Practice Q7(b))
- I can explain each property of definite integrals with a picture or a Riemann sum. (Guide: “The properties of definite integrals”)
- I can explain why a removable discontinuity or a jump does not stop a definite integral from existing. (Guide: “Integrals of functions with jumps and holes”; Practice Q3, Q5(d))
- I can name rules that are not properties, such as splitting a product, and give a counterexample. (Guide: “What is not a property”; Practice Q4)
Skills
- I can evaluate an integral by splitting the region into rectangles, triangles, trapezoids and parts of circles. (Guide: Worked examples 2 and 3; Practice Q2, Q5(a), Q7(a))
- I can recognise √(r² − x²) as a semicircle of radius r. (Guide: “Recognise circles”; Practice Q2)
- I can combine given integral values using reversed limits, constant multiples, sums and adjacent intervals. (Guide: Worked example 1; Practice Q1, Q6(a)–(b))
- I can handle an added constant, using ∫ (a to b) k dx = k(b − a). (Guide: Worked example 1(d); Practice Q5(c), Q6(c))
- I can integrate a piecewise function by splitting at each jump. (Guide: Worked example 2; Practice Q3, Q5)
Reasoning
- I can solve for an unknown constant or limit from a given integral value. (Guide: Worked example 3(b); Practice Q6(d), Q7(d))
- I can tell net area from total area and explain when they differ. (Guide: Worked example 2, Interpretation; Practice Q7(b))
- I can state when these properties do not apply, such as across a vertical asymptote. (Guide: “Boundary”)
All ticked? Move on to Topic 6.7, The Fundamental Theorem of Calculus and Definite Integrals, or return to the Calculus AB roadmap or the Calculus BC roadmap.