When we can’t integrate...

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Presentation transcript:

When we can’t integrate... The Trapezium Rule When we can’t integrate...

Find the shaded area

So can divide this area up into 4 trapeziums of equal width 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 = y0 y1 y2 y3 y4 h h h h ½ (y0 + y1)h + ½ (y1 + y2)h

Total Area = ½ (y0 + y1)h + ½ (y1 + y2)h + ½ (y2 + y3)h + ½ (y3 + y4)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]

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