Riemann Sums

Definition.  Definite Integral as a Limit of a Riemann Sum.   Let    be continuous over the interval  ,  
and let    be a partition,  then the definite integral is given by

    ,  

where    and the mesh size of the partition goes to zero in the "limit,"  i.e .
.   

The following two Mathematica subroutines are used to illustrate this concept, which was introduced in calculus.

The Left Riemann sum uses in the definition.

Mathematica Subroutine (Left Riemann Sum).

The Right Riemann sum uses in the definition.

Mathematica Subroutine (Right Riemann Sum).

The midpoint rule uses in the definition.

Improvements can be made in two directions, the midpoint rule evaluates the function at  ,  which is the midpoint of the subinterval , i.e. in the Riemann sum.  

Mathematica Subroutine (Midpoint Rule).

The Trapezoidal Rule is the average of the left Riemann sum and the right Riemann sum.

Mathematica Subroutine (Trapezoidal Rule).

Example   Let    over  .  Use the left Riemann sum  with  n = 25, 50, and 100  to approximate the value of the integral.