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26. Integration.

26. Integration. a. SERRA. 26A. The definite Integral (of f(x) from a to b). Class work: Investigation - Lower and upper sum of a function. Geogebra and TI-84 {Math>9fnInt( f(x), x,a,b )}.

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26. Integration.

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  1. 26. Integration. a. SERRA

  2. 26A. The definite Integral (of f(x) from a to b) • Class work: • Investigation - Lower and upper sum of a function. • Geogebra 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: • Examples 1 and 2 (link with mock portfolio!): Do them & check your answers. • Exercise 26A: 1,3,5 • Extra: 26A: 2,4,6,8 • NB. If you have extra homework time, use it to review differentiation.

  3. 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: • Exercise 26B: 1,3 • Extra: 26B: 2 and Investigation 1 • NB. If you have extra homework time, use it to review differentiation.

  4. 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: • Copy in your notebook the applications listed in page 656 • Example 4: Do it & check your answers. • Exercise 26C: 1,3 • Extra: 26C: 2,4 • NB. If you have extra homework time, use it to review differentiation.

  5. 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: • Copy in your notebook the properties in page 660 • Example 5: Do it & check your answers. • Exercise 26D: 1 (a,c,e) 2(a,c,e,g) • Extra: 26D: 1 (b,d) 2(b,d,f)

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

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

  8. 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: • Examples 13 and 14: Do them & check your answers. • Exercise 26G: 1,3 • Extra: Exercises 26G (even numbers).

  9. 26H. Distance from velocity • Class work: • Speed vs time function. N.B when the particle reverses direction. • Work an example. • Individual work: • Example 15: Do it & check your answers. • Exercise 26H: 1,3,5 • Extra: Exercises 26H (even numbers).

  10. 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: • Examples 16 and 17: Do them & check your answers. • Exercise 26I: 1 • Extra: Exercises 26I: 2

  11. 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: • Examples 18,19,20 and 21: Do them & check your answers. • Exercise 26J: 1,3,7,11 • Extra: Exercises 26j: even numbers.

  12. 26K. Problem Solving by integration • Class work: • Do the most popular exercise. Otherwise none! • Individual/Pair/Small group work: • Examples 22 and 23: Do them & check your answers. • Exercise 26K: 1,7 • HL (difficult) 26K: 11 (video available) • Extra: Exercises 26K: even numbers.

  13. 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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