BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (limits, L’Hospital’s rule, Topics 6.7 to 6.12) are on the Calculus BC roadmap.
Understanding
- I can say what makes an integral improper: an infinite limit, or an integrand that is unbounded on the interval. (Guide: “The two kinds of improper integral”)
- I can explain what “converges” and “diverges” mean, including divergence by oscillation. (Guide: “The two kinds of improper integral”; Practice Q1)
- I can explain how an infinitely long region can have a finite area. (Guide: “The idea” and Figure 1)
Skills
- I can rewrite an improper integral as a limit of a definite integral, with correct notation. (Guide: definitions table; Practice Q6(a))
- I can evaluate an integral with an infinite limit, using L’Hospital’s rule where needed. (Guide: Worked example 1; Practice Q2, Q6(c))
- I can evaluate an integral whose integrand is unbounded at an endpoint. (Guide: “Benchmarks”; Practice Q3)
- I can split an integral at an asymptote inside the interval and test both parts. (Guide: Worked example 2; Practice Q4)
- I can split an integral from −∞ to ∞ into two halves and evaluate each. (Practice Q5)
- I can state which p-integrals converge on [1, ∞) and on (0, 1]. (Guide: “Benchmarks”; Practice Q1)
Reasoning
- I can show that an integral diverges by finding the limit of the definite integral. (Practice Q6(d), Q7(c))
- I can explain why combining log terms is needed before taking a limit. (Guide: Worked example 1)
- I can interpret a convergent or divergent improper integral as a total in context, with units. (Guide: Worked example 3; Practice Q7)
All ticked? Move on to Topic 6.14, Selecting Techniques for Antidifferentiation, or return to the Calculus BC roadmap.