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 describe a Riemann sum as a total of products, each a function value at a point in a subinterval times that subinterval’s width. (Guide: “What a Riemann sum is, in general” and Figure 1; Practice Q3)
- I can state the definition of the definite integral as a limit of Riemann sums, including the condition that the largest width shrinks to 0. (Guide: “From sums to the definite integral”; Practice Q7(b), Q7(c))
- I can explain why, for a continuous function, the choice of sample points does not change the limit. (Guide: “From sums to the definite integral”; Practice Q6(d))
- I can match each part of ∫ (a to b) f(x) dx with a part of a Riemann sum. (Guide: sum-and-integral table)
Skills
- I can read and evaluate a short sum written in sigma notation, including one whose index starts at 0. (Guide: “Summation notation from first principles”)
- I can write left and right Riemann sums with equal widths in sigma notation, using Δx = (b − a)/n and xᵢ = a + iΔx. (Guide: “Equal widths”; Practice Q5(a), Q5(b))
- I can write a Riemann sum with unequal widths from a table in sigma notation, and calculate it with units. (Practice Q3, Q7(a))
- I can rewrite the limit of a Riemann sum as a definite integral, in more than one correct form. (Guide: Worked example 1; Practice Q1, Q6(a), Q6(b))
- I can rewrite a definite integral as the limit of a right Riemann sum. (Guide: Worked example 2(a); Practice Q2, Q5(c))
Reasoning
- I can spot a wrong translation: a missing width factor, b mistaken for b − a, or a double shift. (Guide: “Common misconceptions”; Practice Q1, Q6(c))
- I can find the value of a limit of sums when the integral is a familiar shape, such as a trapezoid or part of a circle. (Guide: Worked example 2(c); Practice Q4)
- I can say whether a left or right sum overestimates or underestimates, using whether f is increasing or decreasing. (Guide: Worked example 2(c); Practice Q5(d))
All ticked? Move on to Topic 6.4, The Fundamental Theorem of Calculus and Accumulation Functions, or return to the Calculus AB roadmap or the Calculus BC roadmap.