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26. INTEGRATION. A. SERRA

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26A. The definite Integral (of f(x) from a to b) Class work: Investigation - Lower and upper sum of a function. GeogebraGeogebra and TI-84 {Math>9fnInt( f(x), x,a,b)}. Distances from speed-time graphs. For constant speed, for accelerating and decelerating (constant a). Individual work: 1.Examples 1 and 2 (link with mock portfolio!): Do them & check your answers. 2.Exercise 26A: 1,3,5 3.Extra: 26A: 2,4,6,8 4.NB. If you have extra homework time, use it to review differentiation.

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26B. The area function Class work: is the area between the curve y=f(x) and the x-axis between x=a and x=b Consider the function Investigation - Whiteboard (k and x). Discuss relationship between the original functions and the areas…. Individual work: 1.Exercise 26B: 1,3 2.Extra: 26B: 2 and Investigation 1 3.NB. If you have extra homework time, use it to review differentiation.

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26C. Antidifferentiation Class work: If F(x) is such that F’(x) = f(x) (the derivative of F is f) then we say that the antiderivative of f is F and as we discovered in section B is useful to calculate the area under a curve. Read other applications listed in page 656. Brainstorm possible antiderivatives of {3,x,x^2} Individual work: 1.Copy in your notebook the applications listed in page 656 2.Example 4: Do it & check your answers. 3.Exercise 26C: 1,3 4.Extra: 26C: 2,4 5.NB. If you have extra homework time, use it to review differentiation.

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26D. The fundamental theorem of Calculus Class work: State the Fundamental Theorem of calculus: For a continuous function f(x) with antiderivative F(x) In pairs: Use this to prove one of the properties in page 660 (IB exam qns on this). Extra: Prove the fundamental theorem (page 659) Individual work: 1.Copy in your notebook the properties in page 660 2.Example 5: Do it & check your answers. 3.Exercise 26D: 1 (a,c,e) 2(a,c,e,g) 4.Extra: 26D: 1 (b,d) 2(b,d,f)

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26E. Integration Class work: State the definition of integral. Use this to prove the properties in pages 662 and 665 Individual work: 1.Copy in your notebook the properties in pages 662 and 665 2.Examples 6,7,8,9 and 10: Do them & check your answers. 3.Exercise 26E1: 1,5,9 4.Exercise 26E2: 1,5 5.Extra: Exercises 26E1 and 26E2 (even numbers).

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26F. Integrating e^(ax+b) and (ax+b)^n Class work: (Optional) Prove the formulae in page 668 Individual work: 1.Copy formulae in page 668 2.Examples 11and 12: Do them & check your answers. 3.Exercise 26F: 1,5,7 4.Extra: Exercises 26F (even numbers).

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26G. Integrating f(u) u’(x) by substitution Class work: Revise chain rule with the example in page 670. Integrating by substitution vs. chain rule. Individual work: 1.Examples 13 and 14: Do them & check your answers. 2.Exercise 26G: 1,3 3.Extra: Exercises 26G (even numbers).

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26H. Distance from velocity Class work: Speed vs time function. N.B when the particle reverses direction. Work an example. Individual work: 1.Example 15: Do it & check your answers. 2.Exercise 26H: 1,3,5 3.Extra: Exercises 26H (even numbers).

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26I. Definite Integrals using GDC Class work: Definite integrals using Newton’s theorem and using TI-84 {Math>9 fnInt( f(x), x,a,b)}. Individual work: 1.Examples 16 and 17: Do them & check your answers. 2.Exercise 26I: 1 3.Extra: Exercises 26I: 2

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26J. Finding Areas Class work: Revise area under a curve. NB area below X-axis negative. Area between two functions. NB. Sign F(x)-G(x) Individual work: 1.Examples 18,19,20 and 21: Do them & check your answers. 2.Exercise 26J: 1,3,7,11 3.Extra: Exercises 26j: even numbers.

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26K. Problem Solving by integration Class work: Do the most popular exercise. Otherwise none! Individual/Pair/Small group work: 1.Examples 22 and 23: Do them & check your answers. 2.Exercise 26K: 1,7 3.HL (difficult) 26K: 11 (video available) 4.Extra: Exercises 26K: even numbers.

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Review unit 26 and Quiz! INDIVIDUAL WORK HOMEWORK Review Set 26A. (Do, correct your answers and write down score (total and percentage “%”) Extra: Review Set 26B, 26C Next: Multiple choice Online Quiz (0.25)

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