Integration Revision Notes for Core 2

Notes on integration that I created :)

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  • Created by: George
  • Created on: 13-05-11 17:00
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Mathematics ­ Core 2 Integration
Definite and Indefinite integration
Integrating functions without defined limits is called indefinite integration.
(3x2) dx = 3x 2+1/ 3
= x 3+ c (where c is constant)
Integrating functions with defined limits is called definite integration.
If however the question was worded different:
(3x2) dx
= [x 3]
= (2 3) ­ (1 3) = 8 ­ 1
= 7
The three stages of a definite integral
. . . dx = [...] = (. . .) ­ ( . . . )
Definite Integration is defined as
f`(x) dx = [f(x)] = f(b) ­ f(a)

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Example
(2x ­ 3x ½ +1) dx
This means integrate 2x ­ 3x ½ +1 with respect to x between the limits 1 and 4.…read more

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Question
Find the area of the region R bounded by the curve with equation y = (4 ­ x)(x + 2) and the
positive xaxis and yaxis.…read more

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Example
Curve, y = x(4 ­ x)
Line, y = x
Find the area of the region bounded by the curve and the line
Firstly work out the x coordinates of the intersection between the two lines so that we know
what the limits are
x(4 ­ x) = x
4x ­ x 2 = x
x 2 ­ 3x = 0
x(x ­ 3) = 0
x = 0 and x = 3
(4x ­ x 2 ­ x) dx (all I have done…read more

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Alternatively you could have integrated both terms and worked out the area that bounds them to
the x axis between the limits 3 and 0.
So you could have worked out the area of the triangle (shaded blue) and then the area
underneath the curve (shaded blue and red)
Once you have found the area of the triangle subtract it from the area underneath the curve.…read more

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½ (0.5) [1.732 + 2(2 + 2.236 + 2.449) + 2.646]
= ¼ [17.748]
= 4.437
Overestimate and Underestimate
Using the trapezium rule only gives a value for the estimate of the area because.…read more

Comments

E.H Jane

amazing :) 

useful for additional maths (GCSE) aswell :) 

Thank you  so much :D

daviesg

A superb set of revision notes on integration up to core 2. Well presented and supported by excellent examples.

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