Principles of Calculus · SHS2 Additional…

Lesson notes

Learning Objectives

Indicator: 2.3.1.LI.4

By the end of the lesson, learners can:

Explain, in their own words, how increasing the number of subintervals (decreasing step size) affects the accuracy of an estimated area under a curve.
Compute the estimated area under a given curve for at least three different numbers of subintervals using the rectangle (or trapezoid) rule.
Compare the area estimates obtained for different step sizes and state which estimate is closest to the exact area.
Judge, from a table of estimates, whether increasing the number of subintervals always improves the approximation, and justify their reasoning.
Predict the value of the definite integral by observing the trend in successive area estimates as the number of subintervals grows.

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Curriculum details

Strand: Calculus (Strand 3)
Sub-strand: Principles of Calculus (3.1)
Content standard: 2.3.1.CS.2 - Demonstrate a conceptual understanding of the connection between integration and limits and integration as a reverse process of differentiation. 2.3.1.LO.1 Determine the appropriate rule and use it to find the derivative of a function. 2.3.1.LO.2 Estimate the area under a curve using the trapezoid rule.
Indicator: 2.3.1.LI.4 - Compare and judge the effect of reduction in step size and increase in the number of subintervals in an interval for a given function on the area under a curve.
Suggested placement: Semester 2, Week 12 (Week 32 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.394: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 394

Exemplars (from the NaCCA curriculum)

Talk for Learning, Think-pair-share, Experiential Learning and Group Work/Collaborative Learning.
Activity 1: Area of rectangles constructed for different numbers of subintervals In mixed-ability groups, learners find the sum of the area of rectangles constructed for different numbers of subintervals or step sizes on a close interval of a function. Learners record their findings and present them to the whole class for discussion.
Using 𝑓(𝑥) = 𝑥 ) on the interval 0 ≤ 𝑥 ≤ 3, find "
- The definite integral ∫ ( 𝑓(𝑥)𝑑
- On the interval [0,3], get three subintervals with their corresponding rectangles and estimate the integral by summing the areas. Repeat the process 10, 15 and 20 subintervals.
Teaching and Learning Resources:
- Reading resource colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools
Assessment (2.3.1.AS.4). The document marks no depth-of-knowledge level for this indicator.