(*
*
* Copyright (c) 1995, 1996 by Jack Courtney
* All rights reserved.
*
*)
ClearAll[ LHSum, RHSum, Trap, MidPt ];LHSum[ f_, over_List , n_ ]:= Module[ {a=over[[1]],b=over[[2]],h} , h=(b-a)/n ; Return[ Sum[ f[ a+h*i ], {i,0,n-1} ]*h ] ];RHSum[ f_, over_List, n_ ]:= Module[ {a=over[[1]],b=over[[2]],h} , h=(b-a)/n ; Return[ Sum[ f[ a+h*i ], {i,1,n} ]*h ] ];Trap[ f_, over_List, n_ ]:= Module[ {a=over[[1]],b=over[[2]],h} , h=(b-a)/n ; Return[ (Sum[ f[ a+h*i ], {i,1,n-1} ]+(f[a]+f[b])/2)*h ] ];MidPt[ f_, over_List, n_ ]:= Module[ {a=over[[1]],b=over[[2]],h} , h=(b-a)/n ; Return[ h*Sum[ f[ a+h*(i-1/2) ], {i,1,n} ] ] ];(* Sample: (Uncomment to use) *)(*n=10;a=-6.5;b= 6.5;f[x_]=Sin[ (x+6)*Pi/17 ];Print[ "Check Values for Numerical Integration:" ]Print[ "" ];Print[ " Integration of ", OutputForm[ f[x] ], " on (",a,", ",b"):" ];Print[ "" ];Print[ " Exact Value : ", NIntegrate[ f[x],{x,-6.5,6.5} ] ];Print[ " LH Sum (n = ",n,") : ", LHSum[ f, {-6.5,6.5}, n ] //N ];Print[ " RH Sum (n = ",n,") : ", RHSum[ f, {-6.5,6.5}, n ] //N ];Print[ " Trapezoidal Sum (n = ",n,"): ", t=Trap [ f, {-6.5,6.5}, n ] //N ];Print[ " Midpoint Sum (n = ",n,") : ", m=MidPt[ f, {-6.5,6.5}, n ] //N ];Print[ " Simpson Sum (n = ",n,") : ", (2*m+t)/3 //N ];*)
(* Sample output:
Check Values for Numerical Integration:
Integration of Sin[Pi (6 + x)/17] on (-6.5, 6.5 ):
Exact Value : 9.03373
LH Sum (n = 10) : 8.44991
RH Sum (n = 10) : 9.53057
Trapezoidal Sum (n = 10): 8.99024
Midpoint Sum (n = 10) : 9.05549
Simpson Sum (n = 10) : 9.03374
*)