Homework...

• Luke set the homework

Integration and Areas II

Aims: To be able to find the area enclosed between two curves.

To be able to evaluate an integral to infinity ∞

The Area Between A Line and a Curve

• A kind example with give you the points of intersection. You have two choices...

• 1: Find the area below theline between x=1 and x=3.Or 1: Find the area of the trapezium/triangle below theline2: Then find the area belowthe curve between x=1 and x=3.3: Find the difference.

The Area Between A Line and a Curve

-1 1 2 3 4 5 6

-1

1

2

3

4

5

6

x

y

f(x)=–x²+2x+4

g(x)=3x²–6x+4

The Harder Type of Problem

• Here are the curves y=f(x) and y=g(x) where

f(x) = -x2+2x+4g(x) = 3x2-6x+4What is the areaenclosed betweenthem?

What is this area?

The Method• There are two

methods available to find the area.

• The most obvious is to use two integrals to evaluate the two areas and then find the difference.

• There is however an easier method!

-1 1 2 3 4 5 6

-1

1

2

3

4

5

6

x

y

f(x)=–x²+2x+4

g(x)=3x²–6x+4

-1 1 2 3 4 5 6

-1

1

2

3

4

5

6

x

y

f(x)=–x²+2x+4

g(x)=3x²–6x+4

-1 1 2 3 4 5 6

-1

1

2

3

4

5

6

x

y

y=–4x²+8x

y=0

A Quicker WayCalling the higher/larger function f(x) and the smaller function g(x) the rules of integration show that the area between (f(x)-g(x)) and the x-axis has the same value as the area between the two curves over any interval in x.

This Area

Is equal to this one.

y=f(x)-g(x)

Quicker Way

• In general the area enclosed between two curves y=f(x) and y=g(x) between two x values a and b is given by…

dxxgxfb

a )()(

dxFunction

Smaller

Function

Biggerb

a

Solving the Problem: What is the area enclosed by the curves

y=3x2-6x+4 and y=-x2+2x+4

• The Integral is…

2

0

22 )463()42( dxxxxxx coordinates of points of intersection

Higher Function Lower Function

2

0

2 84

tosimplifieswhich

dxxx

Solving the Problem: What is the area enclosed by the curves

y=3x2-6x+4 and y=-x2+2x+4

• So we now integrate and evaluate.

2

0

2 84 dxxx 2334 4

2

0 xx

233423

34 040242

0 3

48332

3

16

Pirate Sail Competition

• Which Pirate Wins...• Each Pirate has a “Sail” graph area.• They score according to this system• A = Area of shaded sail• C = Coolness (product of the highest

powers of x in the two functions)• Final score = AC

Results1st Ching Shape: Area 27/2 -- 54 pts

2nd Black Surder: Area 48/5 -- 38.4pts

3rd Sir Fractal Cube: Area 45/4 -- 33.75pts

4th Squarebeard: Area 9/2 -- 9 pts

5th Long John Circular: 0pts (disqualified for only using one function)

A problem for you…

• Pi-rate Captain Calculus wants a new sail. He visits Sail maker Surd and says “Arrr I be tired of rectangularrr and triangularrr sails I be wantin’ a sail that is the shape of the area enclosed by the graphs of y=x2+2x+2 and y=-x2+2x+10”. Sail maker Surd replied “No problem but what area of sailcloth do I need?”

• Can you work it out?

πPi-rate

Okay in sensible talk: What is the area enclosed by the curves

y=x2+2x+2 and y=-x2+2x+10

1) Identify POIx2+2x+2=-x2+2x+10 2x2-8=0 (2x-4)(x+2)=0 x=2 or x=-22) Identify higher functionWhen x=0 y=x2+2x+2 gives y=2When x=0 y=-x2+2x+10 gives y=10So y=-x2+2x+10 is higher function.3) Integral to solve

2

2

2

2

2

22

)82(

)22()102(

dxx

dxxxxx

-3 -2 -1 1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

10

11

x

y

Okay in sensible talk: What is the area enclosed by the curves

y=x2+2x+2 and y=-x2+2x+10

4) Integrate and evaluate

3

64

332

332

348

316

348

316

3323

32

2

2

2

2

2

28)2(282

332

)82(

8

xx

dxx