Finding Areas Numerically

y4y4 h y5y5 The basic idea is to divide the x-axis into equally spaced divisions as shown and to complete the top of these strips of area in some way so that we can calculate the area by adding up these strips. y1y1 h y2y2 h h y3y3 a b

The first way is to complete the strips as shown. We are then adding up rectangles of height and width. In this case the area is Note that the last y-value is not used. y3y3 h y1y1 h y2y2 h y4y4 y5y5 h ab

y5y5 h y4y4 hh y3y3 y1y1 h y2y2 ab The Trapezium rule. Complete the strips to get trapezia. Add up areas of the form

Simpson ’ s Rule We complete the tops of the strips as shown with parabolas. y5y5 y4y4 y3y3 y1y1 y2y2 ab

x y x Example Rectangle Trapezium Simpson

Given the following table of values, find the approximate value of using Simpson’s rule with 8 sub- intervals.

Given the following table of values, find the approximate value of using Simpson’s rule with 8 sub- intervals.

Simpson’s Rule x y