1. Useful results from chapter 3
Integration
Useful results from chapter 3
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2. Integration
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3. Integration
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4. Integration
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Exercise 4A

5. Integration
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Idea Ex

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

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

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

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

13. Integration by substitution
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Exercise 4C

14. General quadratic denominators
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15. General quadratic denominators
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16. General quadratic denominators
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17. General quadratic denominators
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Exercise 4D

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

19. Integrating inverse trig / hyperbolic functions
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Exercise 4E

20. Reduction Formulae
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Def.
Reduction Formula

21. Reduction Formulae
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22. Reduction Formulae
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23. Reduction Formulae
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24. Reduction Formulae
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Exercise 4F

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

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

27. Arcs and Surfaces
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28. Arcs and Surfaces
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Exercise 4G

29. Arcs and Surfaces
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When a curve is rotated 2 about a line it traces the surface of a solid of revolution a 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.

30. Arcs and Surfaces
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curved
surface
area P

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

33. Arcs and Surfaces
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34. Arcs and Surfaces
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35. Arcs and Surfaces
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Exercise 4H

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

37. + c has been omitted C4

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