Instead of approximating the area under a curve using rectangles or trapezoids, parabolas can be used to approximate each part of a curve.
Let us compute the area under a parabola of the equation passing through the three points :
|
|
|
|
|
|
|
|
|
|
As the points are on the parabola, we have:
|
|
|
|
|
|
|
|
Hence:
|
|
|
|
|
|
Therefore, the area under a parabola can be written as:
|
|
|
|
|
|
Hence by adding all the areas under each parabolic arc using three points we can derive:
The above equation can be simplified into Simpson's rule: