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The Trapezium Rule PowerPoint PPT Presentation


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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

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

a and b are the parallel sides

h is the width


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 =

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 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

= 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 RULE


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