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 say what an antiderivative is and check a proposed one by differentiating it. (Guide: “Antiderivatives”; Practice Q1, Q2)
- I can explain why ∫ (a to x) f(t) dt is an antiderivative of a continuous f, and why two antiderivatives of f differ by a constant. (Guide: “Where the theorem comes from”; Practice Q3)
- I can state the evaluation form of the Fundamental Theorem of Calculus, including the continuity condition. (Guide: “Where the theorem comes from”; Practice Q5(b))
- I can explain why no + C is needed in F(b) − F(a). (Guide: FAQ and “Where the theorem comes from”)
Skills
- I can evaluate a definite integral of powers and roots, rewriting them as powers first. (Guide: Worked example 1; Practice Q1, Q6(a))
- I can evaluate definite integrals of sin x, cos x, sec²x, eˣ and 1/x, getting the trig signs right. (Guide: Worked example 2; Practice Q2, Q5(c))
- I can write [F(x)] (a to b) and subtract F(a) in brackets. (Guide: “How to evaluate a definite integral”; Practice Q2)
- I can turn a limit of Riemann sums into a definite integral and evaluate it. (Guide: “Reading a limit of Riemann sums”; Practice Q4)
Reasoning
- I can spot when the integrand is not continuous on [a, b], and explain why the theorem then gives a wrong answer. (Guide: “When the theorem does not apply”; Practice Q5)
- I can explain a zero or negative definite integral using signed areas. (Guide: Worked example 2(a); Practice Q6)
- I can use ∫ (a to b) F′(x) dx = F(b) − F(a) to find a net change and a final amount, with units. (Guide: Worked example 3; Practice Q7)
All ticked? Move on to Topic 6.8, Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation, or return to the Calculus AB roadmap or the Calculus BC roadmap.