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Integration

Integration. Useful results from chapter 3. Integration. Ex. Integration. Ex. Integration. Ex. Exercise 4A. Ex. Integration. Idea. Ex. Integration. Idea. Remember that identities can help make integrands more familiar…. Ex. Integration. Idea.

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Integration

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  1. Integration Useful results from chapter 3

  2. Integration Ex

  3. Integration Ex

  4. Integration Ex Exercise 4A

  5. Ex Integration Idea

  6. Ex Integration Idea Remember that identities can help make integrands more familiar…

  7. Ex Integration Idea …and don’t forget that hyperbolic functions have easy-to-integrate exponential definitions. Exercise 4B

  8. Integration by substitution Idea These results can be obtained through integration with an appropriate substitution.

  9. Integration by substitution Ex

  10. Integration by substitution Ex

  11. Integration by substitution Ex

  12. Integration by substitution Idea Hyperbolic substitutions can be used more generally Ex

  13. Integration by substitution Ex Exercise 4C

  14. Ex General quadratic denominators ?

  15. General quadratic denominators Ex

  16. General quadratic denominators Ex

  17. General quadratic denominators Ex Exercise 4D

  18. Integrating inverse trig / hyperbolic functions Inverse trig / hyperbolic functions have been appearing as integrals. But how do we do we integrate them? ? Ex

  19. Integrating inverse trig / hyperbolic functions Ex Exercise 4E

  20. Reduction Formula Def. Reduction Formulae ?

  21. Reduction Formulae Ex

  22. Reduction Formulae Ex

  23. Reduction Formulae Ex

  24. Reduction Formulae Ex Exercise 4F

  25. P’ P P’ P (x, y) The rate at which the length of the curve is changing with respect to x Arcs and Surfaces How do find the length of a curve? ?

  26. Arcs and Surfaces How do find the length of a curve? ? Res.

  27. Arcs and Surfaces Ex

  28. Arcs and Surfaces Ex Exercise 4G

  29. a Idea Since we now know how to find lengths of curves. We should be able to adapt the solid of revolution idea to find surfaces of revolution. Arcs and Surfaces When a curve is rotated 2 about a line it traces the surface of a solid of revolution a

  30. P Arcs and Surfaces P’ P curved surface area

  31. Arcs and Surfaces P’ P P

  32. Arcs and Surfaces Res. In practise we will use

  33. Arcs and Surfaces Ex

  34. Arcs and Surfaces Ex

  35. Arcs and Surfaces Ex Exercise 4H

  36. Labels M1 Reference to previous module 1 ? Quick Question Def. Definition Idea Key Idea Ex Example Ex Exercise

  37. C4 + chas been omitted

  38. Arc Length Surface area of revolution + chas been omitted a > 0 where relevant FP3

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