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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…

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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.

So integrating gives


= [x 2 ­ 2x 3/2 + x ]

= (4 2 ­ 2(4) 3/2 + 4) ­ (1 2 ­ 2(1)…

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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.

y = (4 ­ x)(x + 2)

If we plug x as 0 in we get: (4 x 2) = 8

If we…

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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…

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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…

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y = (2x + 1.732 2 2.236 2.449 2.646
3)


= 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.

Comments

E.H Jane

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amazing :) 

useful for additional maths (GCSE) aswell :) 

Thank you  so much :D

daviesg

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

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