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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.
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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
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x
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f(x)=–x²+2x+4
g(x)=3x²–6x+4
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y
f(x)=–x²+2x+4
g(x)=3x²–6x+4
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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
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