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

2. Find the shaded area

3. 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

4. Area of a Trapezium Area = ½ (a + b) h a and b are the parallel sides
h is the width

5. 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

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

7. 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]

8. 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 + y yn-1 ) + yn]

9. TRAPEZIUM RULE