P2 Chapter 8 Integration
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Recap
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Definite Integration
We could add together the area of individual strips, which we want to make as thin as possible…
Definite Integration
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Definite Integration
Reflecting on above, do you think the following definite integrals would be positive or negative or 0?
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0
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✔
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+
0
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✔
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+
0
Evaluating Definite Integrals
We use square brackets to say that we’ve integrated the function, but we’re yet to involve the limits 1 and 2.
Then we find the difference when we sub in our limits.
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Evaluating Definite Integrals
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Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.
Exercise 8B
1
2
4
6
a
c
e
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Harder Examples
Sketch:
(Hint: factorise!)
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Harder Examples
-3
1
The Sketch
The number crunching
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Exercise 8C
1
2
3
4
5
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Curves bound between two lines
Curves bound between two lines
How could we use a similar principle if we were looking for the area bound between two lines?
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therefore area…
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Curves bound between two lines
Bro Tip: We’ll need to find the points at which they intersect.
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Curves bound between two lines
Edexcel C2 May 2013 (Retracted)
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y = x(x-3)
y = 2x
A
B
C
More complex areas
Bro Tip: Sometimes we can subtract areas from others. e.g. Here we could start with the area of the triangle OBC.
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Exercise 8D
1
3
4
9
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Exercise 8D
(Probably more difficult than you’d see in an exam paper, but you never know…)
Q6
7
7
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y1
y2
y3
y4
h
h
h
Trapezium Rule
Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve.
What is the area here?
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Trapezium Rule
In general:
width of each trapezium
Area under curve
is approximately
x | 1 | 1.5 | 2 | 2.5 | 3 |
y | 1 | 2.25 | 4 | 6.25 | 9 |
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Example
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0.8571
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Trapezium Rule
May 2013 (Retracted)
To add: When do we underestimate and overestimate?