The Trapezium Rule

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The Trapezium Rule

• When we cant integrate...

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Find the shaded area
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We dont know how to integrate this function, so
we can use trapeziums to make an estimate
So can divide this area up into 4 trapeziums of
equal width
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Area of a Trapezium
Area ½ (a b) h
a and b are the parallel sides h is the width
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How do we find the height of each side of the
trapeziums?
The height of each trapezium can be found by
substituting the x value into the function to get
y
y2
y3
y4
y1
y0
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Total Area
y2
y0
y1
y3
y4
h
h
h
h
½ (y0 y1)h
½ (y1 y2)h
½ (y2 y3)h
½ (y3 y4)h
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Total Area
½ (y0 y1)h
½ h (y0 y1) (y1 y2) (y2 y3) (y3
y4)
½ h y0 y1 y1 y2 y2 y3 y3 y4
½ h y0 2(y1 y2 y3 ) y4
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TRAPEZIUM RULE
½ h y0 2(y1 y2 y3 ) y4
In general, for any area divided up into n
trapezia of equal width
½ h y0 2(y1 y2 ... yn-1 ) yn
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TRAPEZIUM RULE