Principles of Calculus · SHS2 Additional…

Lesson notes

Learning Objectives

Indicator: 2.3.1.LI.3

By the end of the lesson, learners can:

Define an anti-derivative and explain the role of the constant of integration, C, in indefinite integrals.
Distinguish clearly between definite and indefinite integrals, identifying the lower limit, upper limit, and integrand in definite integral notation.
Write a definite integral in the form ∫ₐᵇ f(x) dx and interpret it as a number representing the area under the curve f(x) from x = a to x = b.
Describe the connection between the limit of the partial sum of areas of rectangles and the definite integral.
Apply the Fundamental Theorem of Calculus to evaluate a definite integral, recognising integration as the reverse process of differentiation.

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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.3 - Identify and write a definite integral notation and its connection to the limits of the partial sum of areas. Identify integration as a reverse process of differentiation.
Suggested placement: Semester 2, Week 11 (Week 31 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.392: 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. 392

Exemplars (from the NaCCA curriculum)

Talk for Learning, Think-pair-share, Experiential Learning and Group Work/Collaborative Learning.
Activity 1: Indefinite and definite integrals In mixed-ability groups, learners distinguish between indefinite and definite integrals and discuss the definition of anti-derivative.
- Definition of anti-derivative An anti-derivative, 𝐹(𝑥), of a function, 𝑓(𝑥), can be defined as a function that can be differentiated to obtain the original function, 𝑓(𝑥). Mathematically,
∫ 𝑓(𝑥)𝑑 = 𝐹(𝑥) + 𝐶, where the derivative of 𝐹(𝑥) is 𝑓(𝑥). i.e., 𝐹'(𝑥) = 𝑓(𝑥) and C is the integration constant
- Definite and indefinite Integrals Given 𝑓(𝑥), a definite integral is a number that represents the area under the curve 𝑓(𝑥) from 𝑥 = 𝑎 𝑡 𝑥 = 𝑏.
Mathematically,
- A definite integral is denoted by
04' ∫ 04& 𝑓(𝑥)𝑑 Where 𝑥 = 𝑎 is the lower limit, and 𝑥 = 𝑏 is the upper limit.
- An indefinite integral is a function denoted by
î 𝑓(𝑥)𝑑
Example: What is the lower and upper limit of the following integrals? )
- ∫ ( (𝑥 ) + 1)𝑑
- 7 (𝑥 ) − 8𝑥)𝑑 ∫ *"
Activity 2: Sum of areas of rectangles Using think pair share, learners are to compare the sum of areas of rectangles constructed as the number of subintervals gets larger (or step size gets to zero) and the definite integral.
% ' ℎ∗t 𝑓(𝑎 + 𝑗ℎ)] ≅ î 𝑓(𝑥)𝑑 & Y4!
Or ' î 𝑓(𝑥) 𝑑 ≅ [𝑓(𝑥 ! )∆𝑥 + 𝑓(𝑥 ) )∆𝑥 + ⋯ + 𝑓(𝑥 % )∆𝑥] & '*& Where, ∆𝑥 = , % 𝑛 (𝑛 𝑜 𝑠)
Fundamental Theorem of Calculus Suppose that 𝑓(𝑥) is continuous on the interval 𝑎 ≤ 𝑥 ≤ 𝑏, and let 𝐹(𝑥) be an antiderivative of 𝑓(𝑥). ' Then ∫ & 𝑓(𝑥)𝑑 = 𝐹(𝑏) − 𝐹(𝑎) Suppose that 𝑓 is a positive continuous function. To 𝑓 we associate an area function 𝐹 that is defined by 0 𝐹(𝑥) = ∫ & 𝑓(𝑡) 𝑑 (𝑎 < 𝑥 < 𝑏)
Understand integration as an inverse process of differentiation
Teaching and Learning Resources:
- Reading resource colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools
Assessment (2.3.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 4 Extended critical thinking and reasoning.