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# The Trapezium Rule PowerPoint PPT Presentation

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.

The Trapezium Rule

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## The Trapezium Rule

When we can’t integrate...

### 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

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]