e.g. Use 4 strips with the mid-ordinate rule to estimate the value of Give the answer to 4 d.p. Solution: We need 4 corresponding y -values for x 1, x 2, x 3,and x 4. x1x1 x2x2 x3x3 x4x4 The Mid Ordinate Rule

0 1 So, a b N.B.

 The number of x-values is the same as the number of strips. SUMMARY where n is the number of strips.  The width, h, of each strip is given by ( but should be checked on a sketch )  The mid-ordinate law for estimating an area is  The accuracy can be improved by increasing n.  The 1st x -value is at the mid-point of the width of the 1 st rectangle:

As before, the area under the curve is divided into a number of strips of equal width. A very good approximation to a definite integral can be found with Simpson’s rule. However, this time, there must be an even number of strips as they are taken in pairs. Simpson’s Rule

SUMMARY where n is the number of strips and must be even.  The width, h, of each strip is given by  Simpson’s rule for estimating an area is  The accuracy can be improved by increasing n.  The number of ordinates ( y -values ) is odd.  a is the left-hand limit of integration and the 1 st value of x.

Use Simpson’s rule with 4 strips to estimate giving your answer to 4 d.p. Solution: ( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. ) Have a go:

Solution:

The following sketches show sample rectangles where the mid-ordinate rule under- and over estimates the area. Underestimates ( concave upwards ) Overestimates ( concave downwards ) How good is your approximation?

Percentage Error Step 1:Use calculator to find definite integral (area).Use calculator Step 2:Apply percentage error formula. Approximation (from Simpson’s Rule or Mid Ordinate Rule minus value of definite integral from calculator divided by value of definite integral from calculator then multiply result by a hundred for percentage. approximation – definite integral x 100 definite integral