15.08 Simpson's rule
Thomas Simpson was a British Mathematician in the 18th Century who came up with a way of approximating the area of shapes with unusually shaped edges.
Typically Simpson's Rule is best used when there is at least $1$1 straight edge to a shape. These can be drawn on to help. The important thing to know is that when using Simpson's rule you must split the shape up into an equal number of strips.
Exploration
Use this applet to see an area under a graphed curve being broken into different numbers of strips. Drag the slider to the right to create more strips. What do you notice?
We saw in the applet above that the more strips we create, the more accurately the area under the curve will be measured.
For this shape we would apply Simpson's Rule just once:
To use Simpson's rule we need to substitute into this formula:
$A=\frac{h}{3}\left(d_f+4d_m+d_l\right)$A=h3(df+4dm+dl)
$h=\text{width of each strip}$h=width of each strip
$d_f=\text{first distance}$df=first distance
$d_m=\text{middle distance}$dm=middle distance
$d_l=\text{last distance}$dl=last distance
If there are more than two strips we need to repeatedly apply Simpson's rule.
Practice questions
Question 1
Use Simpson's Rule to approximate the area of the block of land shown, assuming all measurements are in meters.
Question 2
Consider the following diagram.
Use Simpson's Rule to approximate the area of the upper region to the nearest meter.
Use Simpson's Rule to approximate the area of the lower region to the nearest meter.
Hence approximate the area of the entire block of land to the nearest meter.
Question 3
Use Simpson's Rule twice to find the approximate area of the block of land shown to the nearest meter.
