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The Trapezium Rule. When we can’t integrate. Find the shaded area . We don’t 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. Area of a Trapezium. Area = ½ (a + b) h.

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The Trapezium Rule

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The trapezium rule l.jpg

The Trapezium Rule

When we can’t integrate...


Find the shaded area l.jpg

Find the shaded area


We don t know how to integrate this function so we can use trapeziums to make an estimate l.jpg

We don’t 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


Area of a trapezium l.jpg

Area of a Trapezium

Area = ½ (a + b) h

a and b are the parallel sides

h is the width


How do we find the height of each side of the trapeziums l.jpg

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


Total area l.jpg

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


Total area7 l.jpg

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]

+ ½ (y1 + y2)h

+ ½ (y2 + y3)h

+ ½ (y3 + y4)h


Trapezium rule l.jpg

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]


Trapezium rule9 l.jpg

TRAPEZIUM RULE


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