midpt <- function ( f, a, b, m = 100 )

#*****************************************************************************80
#
## midpt estimates an integral using the midpoint rule.
#
#  Licensing:
#
#    Copyright 2016 James P. Howard, II
#
#    The computer code and data files on this web page are distributed under
#    https://opensource.org/licenses/BSD-2-Clause, the BSD-2-Clause license.
#
#  Modified:
#
#    02 March 2020
#
#  Author:
#
#    Original R code by James Howard;
#    Modifications by John Burkardt.
#
#  Reference:
#
#    James Howard,
#    Computational Methods for Numerical Analysis with R,
#    CRC Press, 2017
#    ISBN13: 978-1-4987-2363-3.
#
{
  nwidth <- ( b - a ) / m
  x <- seq ( a, b - nwidth, length.out = m ) + nwidth / 2.0
  y <- f ( x )
  area <- sum ( y ) * abs ( b - a ) / m

  return ( area )
}