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Integration of polynomials is simple and is based on the calculus. \+ Trapezoidal rule is the area under the curve for a first order polyno mial (straight line) that approximates the integrand. [click " } {URLLINK 17 "here" 4 "numericalmethods.eng.usf.edu/mws/gen/07int/mws_g en_int_txt_trapcontinuous.doc" "" }{TEXT 273 29 " for textbook notes] \+ [ click " }{URLLINK 17 "here" 4 "numericalmethods.eng.usf.edu/mws/gen/ 07int/mws_gen_int_ppt_trapcontinuous.ppt" "" }{TEXT 273 57 " for power point presentation]. " }{TEXT 273 0 "" }}} {SECT 0 {PARA 233 "" 0 "" {TEXT 272 16 "Section II: Data" }{TEXT 272 0 "" }}{PARA 230 "" 0 "" {TEXT 269 149 "The following simulation illus trates the Trapezoidal rule of integration. This section is the only s ection where the user interacts with the program." }{TEXT 273 30 " The user enters any function " }{TEXT 275 4 "f(x)" }{TEXT 273 181 " and t he lower and upper limit of the integration. By entering this data, th e program will calculate the exact value of the integral, followed by \+ the results using Trapezoidal rule " }{TEXT 275 14 "n = 1, 2, 3, 4" } {TEXT 273 11 " segments. " }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 8 "restart;" }{MPLTEXT 1 276 0 "" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 14 "Integrand f(x)" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> \+ " 0 "" {MPLTEXT 1 276 23 "f:=x->300*x/(1+exp(x));" }{MPLTEXT 1 276 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+6#%\"xG6\"6$%)operatorG%&a rrowGF(,$*(\"$+$\"\"\"9$F/,&F/F/-%$expG6#F0F/!\"\"F/F(F(F(6#\"\"!" }}} {EXCHG {PARA 234 "" 0 "" {TEXT 273 32 "The lower limit of the integral " }{TEXT 275 1 "a" }{TEXT 273 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 7 "a:=0.0;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"!F&" }}}{EXCHG {PARA 234 "" 0 "" {TEXT 273 32 "The upper limit of the integral " }{TEXT 275 1 "b" }{TEXT 273 0 " " }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 8 "b:=10.0;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"$+\"!\"\"" }}} {EXCHG {PARA 230 "" 0 "" {TEXT 269 134 "This is the end of the user's \+ section. All information must be entered before proceeding to the next section. Re-execute the program." }{TEXT 269 0 "" }}}{PARA 230 "" 0 "" {TEXT 269 0 "" }}}{SECT 0 {PARA 233 "" 0 "" {TEXT 272 44 "Section I II: The exact value of the integral" }{TEXT 272 0 "" }}{PARA 234 "" 0 "" {TEXT 273 92 "In this section, the program will evaluate the Maple \+ value for the integral of the function " }{TEXT 275 4 "f(x)" }{TEXT 273 25 " evaluated at the limits " }{TEXT 275 1 "a" }{TEXT 273 5 " and " }{TEXT 275 1 "b" }{TEXT 273 1 "." }{TEXT 273 0 "" }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 61 "plot(f(x),x=a..b,title=\"f(x) vs x\", thickness=3, color=black);" }{MPLTEXT 1 276 0 "" }{MPLTEXT 1 276 27 " \ns_exact:=int(f(x),x=a..b);" }{MPLTEXT 1 276 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 457 288 288 {PLOTDATA 2 "6(-%'CURVESG6#7ao7$$\"\"!F)F(7$$\"3 WmmmT&)G\\a!#>$\"3[l?=:tF^z!#<7$$\"3GLLL3x&)*3\"!#=$\"3k=(\\))R!zX:!#; 7$$\"3$*****\\ilyM;F4$\"39X>:@b=_AF77$$\"3emmm;arz@F4$\"3>PwE^.k9HF77$ $\"3;L$e*)4bQl#F4$\"3UntxX,kbMF77$$\"3v***\\7y%*z7$F4$\"3qU8+(>\"4kRF7 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segment (" }{TEXT 279 5 "n = 1" }{TEXT 278 1 ") " }{TEXT 278 0 "" }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 5 "n:=1; " }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"\" " }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 14 "h[1]:=(b-a)/n;" } {MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"\"$ \"$+\"!\"\"" }}}{EXCHG {PARA 234 "" 0 "" {TEXT 273 29 "The integral of the function " }{TEXT 275 4 "f(x)" }{TEXT 273 6 " from " }{TEXT 275 2 "a " }{TEXT 273 3 "to " }{TEXT 275 1 "b" }{TEXT 273 61 " using the t rapezoidal rule with one segment will be equal to" }{TEXT 273 0 "" }}} {EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 26 "s[1]:=(b-a)*(f(a)+f(b))/ 2;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\" \"\"$\"+0.o4o!#5" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 24 "The approx imate error is" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 18 "E_a[1]:=undefined;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"\"%*undefinedG" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 53 "The absolute approximate relative perc entage error is" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 21 "E_arel[1]:=undefined;" }{MPLTEXT 1 276 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"\"%*undefinedG" }}}} {SECT 0 {PARA 238 "" 0 "" {TEXT 278 14 "Two segments (" }{TEXT 279 5 " n = 2" }{TEXT 278 1 ")" }{TEXT 278 0 "" }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 5 "n:=2;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 14 "h[2]:=(b-a)/n;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"#$\"+++++]!\"*" }}}{EXCHG {PARA 234 "" 0 " " {TEXT 273 29 "The integral of the function " }{TEXT 275 4 "f(x)" } {TEXT 273 6 " from " }{TEXT 275 1 "a" }{TEXT 273 4 " to " }{TEXT 275 1 "b" }{TEXT 273 61 " using the trapezoidal rule with two segment will be equal to" }{TEXT 273 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 42 "s[2]:=(b-a)*(f(a)+2*f(a+h[2])+f(b))/(2*n);" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"#$\"++mo`]!\")" }} }{EXCHG {PARA 230 "" 0 "" {TEXT 269 24 "The approximate error is" } {TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 18 "E_a[2] :=s[2]-s[1];" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%$E_aG6#\"\"#$\"+(z*e&)\\!\")" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 53 "The absolute approximate relative percentage error is" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 32 "E_arel[2]:=abs(E _a[2]/s[2]*100);" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"#$\"+8KDl)*!\")" }}}}{SECT 0 {PARA 238 "" 0 "" {TEXT 278 16 "Three segments (" }{TEXT 279 5 "n = 3" }{TEXT 278 1 ")" }{TEXT 278 0 "" }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 5 "n:=3;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }} }{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 14 "h[3]:=(b-a)/n;" } {MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"$$ \"+LLLLL!\"*" }}}{EXCHG {PARA 234 "" 0 "" {TEXT 273 29 "The integral o f the function " }{TEXT 275 4 "f(x)" }{TEXT 273 6 " from " }{TEXT 275 1 "a" }{TEXT 273 4 " to " }{TEXT 275 1 "b" }{TEXT 273 63 " using the t rapezoidal rule with three segment will be equal to" }{TEXT 273 0 "" } }}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 56 "s[3]:=(b-a)*(f(a)+2*f( a+h[3])+2*f(a+2*h[3])+f(b))/(2*n);" }{MPLTEXT 1 276 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"$$\"++v " 0 "" {MPLTEXT 1 276 18 "E_a[3]:=s[3]-s[2];" } {MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"$$ \"++%)3)H(!\")" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 53 "The absolute approximate relative percentage error is" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 32 "E_arel[3]:=abs(E_a[3]/s[3]*100) ;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6 #\"\"$$\"+=T`3f!\")" }}}}{SECT 0 {PARA 238 "" 0 "" {TEXT 278 15 "Four \+ segments (" }{TEXT 279 5 "n = 4" }{TEXT 278 1 ")" }{TEXT 278 0 "" }} {EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 5 "n:=4;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 14 "h[4]:=(b-a)/n;" }{MPLTEXT 1 276 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"%$\"+++++D!\"*" }}} {EXCHG {PARA 234 "" 0 "" {TEXT 273 29 "The integral of the function " }{TEXT 275 4 "f(x)" }{TEXT 273 6 " from " }{TEXT 275 1 "a" }{TEXT 273 4 " to " }{TEXT 275 1 "b" }{TEXT 273 62 " using the trapezoidal rule w ith four segment will be equal to" }{TEXT 273 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 70 "s[4]:=(b-a)*(f(a)+2*f(a+h[4])+2*f(a+2 *h[4])+2*f(a+3*h[4])+f(b))/(2*n);" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"%$\"+0!>hq\"!\"(" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 24 "The approximate error is" }{TEXT 269 0 "" }} }{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 18 "E_a[4]:=s[4]-s[3];" } {MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"%$ \"*0:%4Z!\"(" }}}{EXCHG {PARA 230 "" 0 "" {TEXT 269 53 "The absolute a pproximate relative percentage error is" }{TEXT 269 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 276 32 "E_arel[4]:=abs(E_a[4]/s[4]*100) ;" }{MPLTEXT 1 276 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6 #\"\"%$\"+_'3.w#!\")" }}}{EXCHG }}}{SECT 0 {PARA 233 "" 0 "" {TEXT 272 10 "References" }{TEXT 272 0 "" }}{PARA 230 "" 0 "" {TEXT 269 159 "[1] Autar Kaw, Michael Keteltas, Holistic Numerical Methods Institute , See http://numericalmethods.eng.usf.edu/mws/gen/07int/mws_gen_int_tx t_trapcontinuous.doc" }{TEXT 269 0 "" }}{PARA 230 "" 0 "" {TEXT 269 0 "" }}}{PARA 239 "" 0 "" {TEXT 280 10 "Disclaimer" }{TEXT 281 1 ":" } {TEXT 282 248 " While every effort has been made to validate the solut ions in this worksheet, University of South Florida and the contributo rs are not responsible for any errors contained and are not liable for any damages resulting from the use of this material." }{TEXT 282 0 " " }}{PARA 240 "" 0 "" {TEXT 216 0 "" }}{PARA 241 "" 0 "" {TEXT 283 0 " " }}{PARA 242 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }