Numerical Integration

1. The Rectangular Rule:

   In freshman calculus, the area under a curve, , is defined to be The Rectangular Rule

   

   where is the number of subintervals used in the approximation, simply amounts to the observation that the right hand side of the formula can be used as an approximation for the left. The assumption being that the bigger the better the appromimation. The realities of computing are such that this may or may not be the case. The Mid-point Rule, and Simpson's Rule, below, will, in general, produce a better approximation with fewer intervals.

   

   

2. The Mid-point Rule:

   

   

   

3. Simpson's Rule:(quadratic approximation)

   

   

The Formula for Computing

pirules uses the formula Briefly, the derivation is as follows.

• 

• The Chain Rule: Let then

  Apply the chain rule to

  then

  and

  or