Useful results from chapter 3 Integration Useful results from chapter 3 31 December 2018

Integration 31 December 2018 Ex

Integration 31 December 2018 Ex

Integration 31 December 2018 Ex Exercise 4A

Integration 31 December 2018 Idea Ex

Integration 31 December 2018 Idea Remember that identities can help make integrands more familiar… Ex

Integration Exercise 4B Ex Idea 31 December 2018 Idea …and don’t forget that hyperbolic functions have easy-to-integrate exponential definitions. Exercise 4B Ex

Integration by substitution 31 December 2018 Idea These results can be obtained through integration with an appropriate substitution.

Integration by substitution 31 December 2018 Ex

Integration by substitution 31 December 2018 Ex

Integration by substitution 31 December 2018 Ex

Integration by substitution 31 December 2018 Idea Hyperbolic substitutions can be used more generally Ex

Integration by substitution 31 December 2018 Ex Exercise 4C

General quadratic denominators 31 December 2018 ? Ex

General quadratic denominators 31 December 2018 Ex

General quadratic denominators 31 December 2018 Ex

General quadratic denominators 31 December 2018 Ex Exercise 4D

Integrating inverse trig / hyperbolic functions 31 December 2018 Inverse trig / hyperbolic functions have been appearing as integrals. But how do we do we integrate them? ? Ex

Integrating inverse trig / hyperbolic functions 31 December 2018 Ex Exercise 4E

Reduction Formulae 31 December 2018 ? Def. Reduction Formula

Reduction Formulae 31 December 2018 Ex

Reduction Formulae 31 December 2018 Ex

Reduction Formulae 31 December 2018 Ex

Reduction Formulae 31 December 2018 Ex Exercise 4F

? Arcs and Surfaces How do find the length of a curve? P’ P P’ P 31 December 2018 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

? Arcs and Surfaces How do find the length of a curve? Res. 31 December 2018 How do find the length of a curve? ? Res.

Arcs and Surfaces 31 December 2018 Ex

Arcs and Surfaces 31 December 2018 Ex Exercise 4G

Arcs and Surfaces 31 December 2018 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.

Arcs and Surfaces 31 December 2018 P’ P curved surface area P

Arcs and Surfaces 31 December 2018 P’ P P

Arcs and Surfaces 31 December 2018 Res. In practise we will use

Arcs and Surfaces 31 December 2018 Ex

Arcs and Surfaces 31 December 2018 Ex

Arcs and Surfaces 31 December 2018 Ex Exercise 4H

? M1 Labels Ex Ex Def. Idea Reference to previous module 1 31 December 2018 M1 Reference to previous module 1 ? Quick Question Def. Definition Idea Key Idea Ex Example Ex Exercise

+ c has been omitted C4

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