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FP3 Chapter 4 Integration

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FP3 Integration Overview

This chapter is long and perilous, but you will learn lots of interesting new techniques as well as reprising existing ones…

Section A: �General Skills

Section B: �Reduction Formulae

A technique for dealing with large powers in integration.

Section C: �Arc lengths and surface area

Surface area of volumes of revolution.

Length of a curve.

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SECTION A PART 1 :: Standard Integrals

Same as non-hyperbolic version?

✔

🗶

Not in formula booklet.

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Quickfire Examples – Do From Memory!

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Click only if you’ve forgotten them.

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Integration by Recognition

Bro Tip: If there’s a power outside in the denominator, always reexpress as product first.

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Exercise 4A

1

2

a

b

c

d

e

f

g

a

b

c

d

e

f

g

3

4

h

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Exercise 4A

5

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Integrating when not quite so standard

Method 1: “Consider and scale”

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Integrating when not quite so standard

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Using Identities

Recap: If you forget a hyperbolic identity, use Osborn’s Rule.

Use this approach in general for small odd powers of sinh and cosh.

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When that doesn’t work…

Sometimes there are techniques which work on non-hyperbolic trig functions but doesn’t work on hyperbolic ones.

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(Bro Exam Note: This very question appeared June 2014, except involving definite integration)

(Integration by parts DOES also work, but requires a significantly greater amount of working!)

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Exercise 4B

1

2

3

4

5

6

7

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Exercise 4B

8

9

10

11

12

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Using substitutions

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Sensible substitution and why?

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Harder Example

(Hint: Use a sensible substitution)

? Using a seemingly-sensible-but-turns-out-rather-nasty substitution

? Using the other-possible-substitution-that-turns-out-much-more-pretty-yay

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Test Your Understanding

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Exercise 4C

Unless a substitution is given or asked for, use the standard results. Give numerical answers to 3 sf.

1

2

3

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Integrating by Completing the Square

These can be obtained using C4 techniques: splitting into partial fractions first.

By completing the square, we can then use one of the standard results.

This is not in the standard form yet, but a simple substitution would make it so.

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Further Example

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Test Your Understanding

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Exercise 4D

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Integration by Parts

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Bro Exam Note: This has never specifically come up in an exam, but could be tested.

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Test Your Understanding

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Exercise 4E

1

2

3

4

5

6

a

b

c

a

b

a

b

c

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Section A: �General Skills

Section B: �Reduction Formulae

A technique for dealing with large powers in integration.

Section C: �Arc lengths and surface area

Surface area of volumes of revolution.

Lengths of a curve.

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Reduction Formulae

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Reduction Formulae

a

b

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Further Example

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A Trig Example

a

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A Trig Example

b

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Test Your Understanding

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Exercise 4F

1

2

3

4

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Exercise 4F

5

6

7

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Exercise 4F

8

9

10

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Section A: �General Skills

Section B: �Reduction Formulae

A technique for dealing with large powers in integration.

Section C: �Arc lengths and surface area

Surface area of volumes of revolution.

Lengths of a curve.

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Lengths of curves

You also used a similar strategy in FP2 to get the area under a polar curve.

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Lengths of curves

So what infinitely small things should we add this time for the length of a curve?

We add together infinitely small straight lines/chords. We can use Pythagoras to get the length of each line.

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Lengths of curves

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Example

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Example

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Test Your Understanding

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Exercise 4G

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Fun Fact

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Fun Fact

Surprisingly, it is not possible to find the exact length of a general ellipse.

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Surface Area of Revolution

It’s the curved surface area of a frustum!

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Surface Area of Revolution

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Surface Area of Revolution

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Example

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Further Example

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Test Your Understanding

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Exercise 4H

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Summary cheat sheet

		Strategy/Notes	FmBk?
			Yes
			Yes
			Yes
		(Same as non-hyp)	No
		(Same as non-hyp)	No
		(NOT same as non-hyp)	No
		(Same as non-hyp)	No
			Yes
			Yes
			Yes
			Yes
		Partial Fractions	Yes
		Partial Fractions	Yes

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Summary cheat sheet

		Strategy/Notes	FmBk?
			No
			No
			No
			No
			No
			No
			No

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