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19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" 
"Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur
ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3
8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "
Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv
e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57
" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "C
urve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve
 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76
" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "C
urve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve
 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95
" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" 
}}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"$+\"\"\"\"-%$SumG6$,&\"
\"*!\"\"*&#F&\"#?F(-%\"iG6#,&\"\"#F.*(F&F(F'F.F3F(F(F(F(/F3;F(F'F(F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++++vU!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 599 "We see that the answer is nowhere close to 11.7 and
 is in fact negative. Looking at the approximating rectangles shows us
 that in the sum when we take the height to be f(x*) for some x* in th
e subinterval , this value will be negative when x is between -2 and .
2. So we guess that this process will be yielding the negative of the \+
area of the first triangle + the area of the second triangle. That is \+
1.8*(-9)*.5 + 1.2*(4).5= -8.1 + 3.6=-4.5. This answer seems more consi
stent with the -4.275 above. This is considered a \"Net Area\" ie Area
 above x axis - Area belowbelow x axis. We switch to n=1000." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "rightbox(f(x),x=-2..1,1000);
 rightsum(f(x),x=-2..1,1000); evalf(%);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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XF^\\cl7$$F[_mF\\]yF\\acl7$F_aclF^[lFc\\l-Fh[l6$7&F`acl7$F_acl$\"+++++
gF^\\cl7$$F_emF\\]yFeacl7$FhaclF^[lFc\\l-Fh[l6$7&Fiacl7$Fhacl$\"+++++v
F^\\cl7$$Fc[nF\\]yF^bcl7$FabclF^[lFc\\l-Fh[l6$7&Fbbcl7$Fabcl$\"+++++!*
F^\\cl7$$FganF\\]yFgbcl7$FjbclF^[lFc\\l-Fh[l6$7&F[ccl7$Fjbcl$\"++++]5F
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F\\]yF\\fcl7$F_fclF^[lFc\\l-Fh[l6$7&F`fcl7$F_fcl$\"++++]>F\\]y7$$Fc]qF
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cl7$FeiclF^[lFc\\l-Fh[l6$7&Fficl7$Feicl$\"++++]GF\\]y7$$F[csF\\]yF[jcl
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^[lFc\\l-Fh[l6$7&Fi[dl7$Fh[dl$\"++++]MF\\]y7$$F[\\uF\\]yF^\\dl7$Fa\\dl
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dlF^[lFc\\l-Fh[l6$7&F[]dl7$Fj\\dl$\"++++]PF\\]y7$$FchuF\\]yF`]dl7$Fc]d
lF^[lFc\\l-Fh[l6$7&Fd]dl7$Fc]dl$\"+++++RF\\]y7$$Fg^vF\\]yFi]dl7$F\\^dl
F^[lFc\\l-Fh[l6$7&F]^dl7$F\\^dl$\"++++]SF\\]y7$$F[evF\\]yFb^dl7$Fe^dlF
^[lFc\\l-Fh[l6$7&Ff^dl7$Fe^dl$\"+++++UF\\]y7$$F_[wF\\]yF[_dl7$F^_dlF^[
lFc\\l-Fh[l6$7&F__dl7$F^_dl$\"++++]VF\\]y7$$FcawF\\]yFd_dl7$Fg_dlF^[lF
c\\l-Fh[l6$7&Fh_dl7$Fg_dl$F]aclF\\]y7$$FggwF\\]yF]`dl7$F_`dlF^[lFc\\l-
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[l6$7&Fi`dl7$Fh`dl$\"+++++[F\\]y7$$F_dxF\\]yF^adl7$FaadlF^[lFc\\l-Fh[l
6$7&Fbadl7$Faadl$\"++++]\\F\\]y7$$FcjxF\\]yFgadl7$FjadlF^[lFc\\l-Fh[l6
$7&F[bdl7$Fjadl$\"+++++^F\\]y7$$Fh`yF^\\clF`bdl7$FcbdlF^[lFc\\l-Fh[l6$
7&Fdbdl7$Fcbdl$\"++++]_F\\]y7$$F\\gyF^\\clFibdl7$F\\cdlF^[lFc\\l-Fh[l6
$7&F]cdl7$F\\cdl$\"+++++aF\\]y7$$F`]zF^\\clFbcdl7$FecdlF^[lFc\\l-Fh[l6
$7&Ffcdl7$Fecdl$\"++++]bF\\]y7$$FdczF^\\clF[ddl7$F^ddlF^[lFc\\l-Fh[l6$
7&F_ddl7$F^ddl$\"+++++dF\\]y7$$FhizF^\\clFdddl7$FgddlF^[lFc\\l-Fh[l6$7
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7&Faedl7$F`edl$FfaclF\\]y7$$F`f[lF^\\clFfedl7$FhedlF^[lFc\\l-Fh[l6$7&F
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7&F[gdl7$Fjfdl$\"++++]kF\\]y7$$F\\i\\lF^\\clF`gdl7$FcgdlF^[lFc\\l-Fh[l
6$7&Fdgdl7$Fcgdl$\"+++++mF\\]y7$$F`_]lF^\\clFigdl7$F\\hdlF^[lFc\\l-Fh[
l6$7&F]hdl7$F\\hdl$\"++++]nF\\]y7$$Fde]lF^\\clFbhdl7$FehdlF^[lFc\\l-Fh
[l6$7&Ffhdl7$Fehdl$\"+++++pF\\]y7$$Fh[^lF^\\clF[idl7$F^idlF^[lFc\\l-Fh
[l6$7&F_idl7$F^idl$\"++++]qF\\]y7$$F\\b^lF^\\clFdidl7$FgidlF^[lFc\\l-F
h[l6$7&Fhidl7$Fgidl$\"+++++sF\\]y7$$F`h^lF^\\clF]jdl7$F`jdlF^[lFc\\l-F
h[l6$7&Fajdl7$F`jdl$\"++++]tF\\]y7$$Fd^_lF^\\clFfjdl7$FijdlF^[lFc\\l-F
h[l6$7&Fjjdl7$Fijdl$F_bclF\\]y7$$Fhd_lF^\\clF_[el7$Fa[elF^[lFc\\l-Fh[l
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l6$7&F[\\el7$Fj[el$\"+++++yF\\]y7$$F`a`lF^\\clF`\\el7$Fc\\elF^[lFc\\l-
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^[lFc\\l-Fh[l6$7&F`iel7$F^iel$\"++++D6F,7$$\"+++++DF^\\clFeiel7$FhielF
^[lFc\\l-Fh[l6$7&Fjiel7$Fhiel$\"++++S6F,7$$\"+++++GF^\\clF_jel7$FbjelF
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$Fb]flF^[lFc\\l-Fh[l6$7&Fd]fl7$Fb]fl$\"++++I7F,7$$\"+++++YF^\\clFi]fl7
$F\\^flF^[lFc\\l-Fh[l6$7&F^^fl7$F\\^fl$\"++++X7F,7$$\"+++++\\F^\\clFc^
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F_efl7$FbeflF^[lFc\\l-Fh[l6$7&Fdefl7$Fbefl$\"++++D9F,7$$\"+++++&)F^\\c
lFiefl7$F\\fflF^[lFc\\l-Fh[l6$7&F^ffl7$F\\ffl$\"++++S9F,7$$\"+++++))F^
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F^\\clF]gfl7$F`gflF^[lFc\\l-Fh[l6$7&Fbgfl7$F`gfl$\"++++q9F,7$$\"+++++%
*F^\\clFggfl7$FjgflF^[lFc\\l-Fh[l6$7&F\\hfl7$Fjgfl$\"++++&[\"F,7$$\"++
+++(*F^\\clFahfl7$FdhflF^[lFc\\l-Fh[l6$7&Ffhfl7$Fdhfl$F[`clF,7$$F[eelF
\\]yF[ifl7$F]iflF^[lFc\\l-Fh[l6$7&F^ifl7$F]ifl$\"++++::F,7$$\"++++I5F
\\]yFcifl7$FfiflF^[lFc\\l-Fh[l6$7&Fhifl7$Ffifl$\"++++I:F,7$$\"++++g5F
\\]yF]jfl7$F`jflF^[lFc\\l-Fh[l6$7&Fbjfl7$F`jfl$\"++++X:F,7$$\"++++!4\"
F\\]yFgjfl7$FjjflF^[lFc\\l-Fh[l6$7&F\\[gl7$Fjjfl$\"++++g:F,7$$\"++++?6
F\\]yFa[gl7$Fd[glF^[lFc\\l-Fh[l6$7&Ff[gl7$Fd[gl$\"++++v:F,7$$\"++++]6F
\\]yF[\\gl7$F^\\glF^[lFc\\l-Fh[l6$7&F`\\gl7$F^\\gl$\"++++!f\"F,7$$\"++
++!=\"F\\]yFe\\gl7$Fh\\glF^[lFc\\l-Fh[l6$7&Fj\\gl7$Fh\\gl$\"++++0;F,7$
$\"++++57F\\]yF_]gl7$Fb]glF^[lFc\\l-Fh[l6$7&Fd]gl7$Fb]gl$\"++++?;F,7$$
\"++++S7F\\]yFi]gl7$F\\^glF^[lFc\\l-Fh[l6$7&F^^gl7$F\\^gl$\"++++N;F,7$
$\"++++q7F\\]yFc^gl7$Ff^glF^[lFc\\l-Fh[l6$7&Fh^gl7$Ff^gl$FdeclF,7$$Fag
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8F\\]yFe_gl7$Fh_glF^[lFc\\l-Fh[l6$7&Fj_gl7$Fh_gl$\"++++!o\"F,7$$\"++++
g8F\\]yF_`gl7$Fb`glF^[lFc\\l-Fh[l6$7&Fd`gl7$Fb`gl$\"++++&p\"F,7$$\"+++
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+++]9F\\]yF]bgl7$F`bglF^[lFc\\l-Fh[l6$7&Fbbgl7$F`bgl$\"++++S<F,7$$\"++
++![\"F\\]yFgbgl7$FjbglF^[lFc\\l-Fh[l6$7&F\\cgl7$Fjbgl$\"++++b<F,7$$\"
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+++S:F\\]yF[dgl7$F^dglF^[lFc\\l-Fh[l6$7&F`dgl7$F^dgl$\"++++&y\"F,7$$\"
++++q:F\\]yFedgl7$FhdglF^[lFc\\l-Fh[l6$7&Fjdgl7$Fhdgl$F]fclF,7$$F[helF
\\]yF_egl7$FaeglF^[lFc\\l-Fh[l6$7&Fbegl7$Faegl$\"++++:=F,7$$\"++++I;F
\\]yFgegl7$FjeglF^[lFc\\l-Fh[l6$7&F\\fgl7$Fjegl$\"++++I=F,7$$\"++++g;F
\\]yFafgl7$FdfglF^[lFc\\l-Fh[l6$7&Fffgl7$Fdfgl$\"++++X=F,7$$\"++++!p\"
F\\]yF[ggl7$F^gglF^[lFc\\l-Fh[l6$7&F`ggl7$F^ggl$\"++++g=F,7$$\"++++?<F
\\]yFeggl7$FhgglF^[lFc\\l-Fh[l6$7&Fjggl7$Fhggl$\"++++v=F,7$$\"++++]<F
\\]yF_hgl7$FbhglF^[lFc\\l-Fh[l6$7&Fdhgl7$Fbhgl$\"++++!*=F,7$$\"++++!y
\"F\\]yFihgl7$F\\iglF^[lFc\\l-Fh[l6$7&F^igl7$F\\igl$\"++++0>F,7$$\"+++
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S=F\\]yF]jgl7$F`jglF^[lFc\\l-Fh[l6$7&Fbjgl7$F`jgl$\"++++N>F,7$$\"++++q
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Fa[hl7$Fc[hlF^[lFc\\l-Fh[l6$7&Fd[hl7$Fc[hl$\"++++l>F,7$$\"++++I>F\\]yF
i[hl7$F\\\\hlF^[lFc\\l-Fh[l6$7&F^\\hl7$F\\\\hl$\"++++!)>F,7$$\"++++g>F
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+]?F\\]yFa^hl7$Fd^hlF^[lFc\\l-Fh[l6$7&Ff^hl7$Fd^hl$\"++++S?F,7$$\"++++
!3#F\\]yF[_hl7$F^_hlF^[lFc\\l-Fh[l6$7&F`_hl7$F^_hl$\"++++b?F,7$$\"++++
5@F\\]yFe_hl7$Fh_hlF^[lFc\\l-Fh[l6$7&Fj_hl7$Fh_hl$\"++++q?F,7$$\"++++S
@F\\]yF_`hl7$Fb`hlF^[lFc\\l-Fh[l6$7&Fd`hl7$Fb`hl$\"++++&3#F,7$$\"++++q
@F\\]yFi`hl7$F\\ahlF^[lFc\\l-Fh[l6$7&F^ahl7$F\\ahl$F_gclF,7$$F_ielF\\]
yFcahl7$FeahlF^[lFc\\l-Fh[l6$7&Ffahl7$Feahl$\"++++:@F,7$$\"++++IAF\\]y
F[bhl7$F^bhlF^[lFc\\l-Fh[l6$7&F`bhl7$F^bhl$\"++++I@F,7$$\"++++gAF\\]yF
ebhl7$FhbhlF^[lFc\\l-Fh[l6$7&Fjbhl7$Fhbhl$\"++++X@F,7$$\"++++!H#F\\]yF
_chl7$FbchlF^[lFc\\l-Fh[l6$7&Fdchl7$Fbchl$\"++++g@F,7$$\"++++?BF\\]yFi
chl7$F\\dhlF^[lFc\\l-Fh[l6$7&F^dhl7$F\\dhl$\"++++v@F,7$$\"++++]BF\\]yF
cdhl7$FfdhlF^[lFc\\l-Fh[l6$7&Fhdhl7$Ffdhl$\"++++!>#F,7$$\"++++!Q#F\\]y
F]ehl7$F`ehlF^[lFc\\l-Fh[l6$7&Fbehl7$F`ehl$\"++++0AF,7$$\"++++5CF\\]yF
gehl7$FjehlF^[lFc\\l-Fh[l6$7&F\\fhl7$Fjehl$\"++++?AF,7$$\"++++SCF\\]yF
afhl7$FdfhlF^[lFc\\l-Fh[l6$7&Fffhl7$Fdfhl$\"++++NAF,7$$\"++++qCF\\]yF[
ghl7$F^ghlF^[lFc\\l-Fh[l6$7&F`ghl7$F^ghl$FhgclF,7$$FiielF\\]yFeghl7$Fg
ghlF^[lFc\\l-Fh[l6$7&Fhghl7$Fgghl$\"++++lAF,7$$\"++++IDF\\]yF]hhl7$F`h
hlF^[lFc\\l-Fh[l6$7&Fbhhl7$F`hhl$\"++++!G#F,7$$\"++++gDF\\]yFghhl7$Fjh
hlF^[lFc\\l-Fh[l6$7&F\\ihl7$Fjhhl$\"++++&H#F,7$$\"++++!f#F\\]yFaihl7$F
dihlF^[lFc\\l-Fh[l6$7&Ffihl7$Fdihl$\"++++5BF,7$$\"++++?EF\\]yF[jhl7$F^
jhlF^[lFc\\l-Fh[l6$7&F`jhl7$F^jhl$\"++++DBF,7$$\"++++]EF\\]yFejhl7$Fhj
hlF^[lFc\\l-Fh[l6$7&Fjjhl7$Fhjhl$\"++++SBF,7$$\"++++!o#F\\]yF_[il7$Fb[
ilF^[lFc\\l-Fh[l6$7&Fd[il7$Fb[il$\"++++bBF,7$$\"++++5FF\\]yFi[il7$F\\
\\ilF^[lFc\\l-Fh[l6$7&F^\\il7$F\\\\il$\"++++qBF,7$$\"++++SFF\\]yFc\\il
7$Ff\\ilF^[lFc\\l-Fh[l6$7&Fh\\il7$Ff\\il$\"++++&Q#F,7$$\"++++qFF\\]yF]
]il7$F`]ilF^[lFc\\l-Fh[l6$7&Fb]il7$F`]il$FahclF,7$$FcjelF\\]yFg]il7$Fi
]ilF^[lFc\\l-Fh[l6$7&Fj]il7$Fi]il$\"++++:CF,7$$\"++++IGF\\]yF_^il7$Fb^
ilF^[lFc\\l-Fh[l6$7&Fd^il7$Fb^il$\"++++ICF,7$$\"++++gGF\\]yFi^il7$F\\_
ilF^[lFc\\l-Fh[l6$7&F^_il7$F\\_il$\"++++XCF,7$$\"++++!*GF\\]yFc_il7$Ff
_ilF^[lFc\\l-Fh[l6$7&Fh_il7$Ff_il$\"++++gCF,7$$\"++++?HF\\]yF]`il7$F``
ilF^[lFc\\l-Fh[l6$7&Fb`il7$F``il$\"++++vCF,7$$\"++++]HF\\]yFg`il7$Fj`i
lF^[lFc\\l-Fh[l6$7&F\\ail7$Fj`il$\"++++!\\#F,7$$\"++++!)HF\\]yFaail7$F
dailF^[lFc\\l-Fh[l6$7&Ffail7$Fdail$\"++++0DF,7$$\"++++5IF\\]yF[bil7$F^
bilF^[lFc\\l-Fh[l6$7&F`bil7$F^bil$\"++++?DF,7$$\"++++SIF\\]yFebil7$Fhb
ilF^[lFc\\l-Fh[l6$7&Fjbil7$Fhbil$\"++++NDF,7$$\"++++qIF\\]yF_cil7$Fbci
lF^[lFc\\l-Fh[l6$7&Fdcil7$Fbcil$FjhclF,7$$F][flF\\]yFicil7$F[dilF^[lFc
\\l-Fh[l6$7&F\\dil7$F[dil$\"++++lDF,7$$\"++++IJF\\]yFadil7$FddilF^[lFc
\\l-Fh[l6$7&Ffdil7$Fddil$\"++++!e#F,7$$\"++++gJF\\]yF[eil7$F^eilF^[lFc
\\l-Fh[l6$7&F`eil7$F^eil$\"++++&f#F,7$$\"++++!>$F\\]yFeeil7$FheilF^[lF
c\\l-Fh[l6$7&Fjeil7$Fheil$\"++++5EF,7$$\"++++?KF\\]yF_fil7$FbfilF^[lFc
\\l-Fh[l6$7&Fdfil7$Fbfil$\"++++DEF,7$$\"++++]KF\\]yFifil7$F\\gilF^[lFc
\\l-Fh[l6$7&F^gil7$F\\gil$\"++++SEF,7$$\"++++!G$F\\]yFcgil7$FfgilF^[lF
c\\l-Fh[l6$7&Fhgil7$Ffgil$\"++++bEF,7$$\"++++5LF\\]yF]hil7$F`hilF^[lFc
\\l-Fh[l6$7&Fbhil7$F`hil$\"++++qEF,7$$\"++++SLF\\]yFghil7$FjhilF^[lFc
\\l-Fh[l6$7&F\\iil7$Fjhil$\"++++&o#F,7$$\"++++qLF\\]yFaiil7$FdiilF^[lF
c\\l-Fh[l6$7&Ffiil7$Fdiil$FciclF,7$$Fg[flF\\]yF[jil7$F]jilF^[lFc\\l-Fh
[l6$7&F^jil7$F]jil$\"++++:FF,7$$\"++++IMF\\]yFcjil7$FfjilF^[lFc\\l-Fh[
l6$7&Fhjil7$Ffjil$\"++++IFF,7$$\"++++gMF\\]yF][jl7$F`[jlF^[lFc\\l-Fh[l
6$7&Fb[jl7$F`[jl$\"++++XFF,7$$\"++++!\\$F\\]yFg[jl7$Fj[jlF^[lFc\\l-Fh[
l6$7&F\\\\jl7$Fj[jl$\"++++gFF,7$$\"++++?NF\\]yFa\\jl7$Fd\\jlF^[lFc\\l-
Fh[l6$7&Ff\\jl7$Fd\\jl$\"++++vFF,7$$\"++++]NF\\]yF[]jl7$F^]jlF^[lFc\\l
-Fh[l6$7&F`]jl7$F^]jl$\"++++!z#F,7$$\"++++!e$F\\]yFe]jl7$Fh]jlF^[lFc\\
l-Fh[l6$7&Fj]jl7$Fh]jl$\"++++0GF,7$$\"++++5OF\\]yF_^jl7$Fb^jlF^[lFc\\l
-Fh[l6$7&Fd^jl7$Fb^jl$\"++++?GF,7$$\"++++SOF\\]yFi^jl7$F\\_jlF^[lFc\\l
-Fh[l6$7&F^_jl7$F\\_jl$\"++++NGF,7$$\"++++qOF\\]yFc_jl7$Ff_jlF^[lFc\\l
-Fh[l6$7&Fh_jl7$Ff_jl$F\\jclF,7$$Fa\\flF\\]yF]`jl7$F_`jlF^[lFc\\l-Fh[l
6$7&F``jl7$F_`jl$\"++++lGF,7$$\"++++IPF\\]yFe`jl7$Fh`jlF^[lFc\\l-Fh[l6
$7&Fj`jl7$Fh`jl$\"++++!)GF,7$$\"++++gPF\\]yF_ajl7$FbajlF^[lFc\\l-Fh[l6
$7&Fdajl7$Fbajl$\"++++&*GF,7$$\"++++!z$F\\]yFiajl7$F\\bjlF^[lFc\\l-Fh[
l6$7&F^bjl7$F\\bjl$\"++++5HF,7$$\"++++?QF\\]yFcbjl7$FfbjlF^[lFc\\l-Fh[
l6$7&Fhbjl7$Ffbjl$\"++++DHF,7$$\"++++]QF\\]yF]cjl7$F`cjlF^[lFc\\l-Fh[l
6$7&Fbcjl7$F`cjl$\"++++SHF,7$$\"++++!)QF\\]yFgcjl7$FjcjlF^[lFc\\l-Fh[l
6$7&F\\djl7$Fjcjl$\"++++bHF,7$$\"++++5RF\\]yFadjl7$FddjlF^[lFc\\l-Fh[l
6$7&Ffdjl7$Fddjl$\"++++qHF,7$$\"++++SRF\\]yF[ejl7$F^ejlF^[lFc\\l-Fh[l6
$7&F`ejl7$F^ejl$\"++++&)HF,7$$\"++++qRF\\]yFeejl7$FhejlF^[lFc\\l-Fh[l6
$7&Fjejl7$Fhejl$\"\"$F*7$$FeeelF\\]yF_fjl7$FbfjlF^[lFc\\l-Fh[l6$7&Fcfj
l7$Fbfjl$\"++++:IF,7$$\"++++ISF\\]yFhfjl7$F[gjlF^[lFc\\l-Fh[l6$7&F]gjl
7$F[gjl$\"++++IIF,7$$\"++++gSF\\]yFbgjl7$FegjlF^[lFc\\l-Fh[l6$7&Fggjl7
$Fegjl$\"++++XIF,7$$\"++++!4%F\\]yF\\hjl7$F_hjlF^[lFc\\l-Fh[l6$7&Fahjl
7$F_hjl$\"++++gIF,7$$\"++++?TF\\]yFfhjl7$FihjlF^[lFc\\l-Fh[l6$7&F[ijl7
$Fihjl$\"++++vIF,7$$\"++++]TF\\]yF`ijl7$FcijlF^[lFc\\l-Fh[l6$7&Feijl7$
Fcijl$\"++++!4$F,7$$\"++++!=%F\\]yFjijl7$F]jjlF^[lFc\\l-Fh[l6$7&F_jjl7
$F]jjl$\"++++0JF,7$$\"++++5UF\\]yFdjjl7$FgjjlF^[lFc\\l-Fh[l6$7&Fijjl7$
Fgjjl$\"++++?JF,7$$\"++++SUF\\]yF^[[m7$Fa[[mF^[lFc\\l-Fh[l6$7&Fc[[m7$F
a[[m$\"++++NJF,7$$\"++++qUF\\]yFh[[m7$F[\\[mF^[lFc\\l-Fh[l6$7&F]\\[m7$
F[\\[m$F][dlF,7$$Fc]flF\\]yFb\\[m7$Fd\\[mF^[lFc\\l-Fh[l6$7&Fe\\[m7$Fd
\\[m$\"++++lJF,7$$\"++++IVF\\]yFj\\[m7$F]][mF^[lFc\\l-Fh[l6$7&F_][m7$F
]][m$\"++++!=$F,7$$\"++++gVF\\]yFd][m7$Fg][mF^[lFc\\l-Fh[l6$7&Fi][m7$F
g][m$\"++++&>$F,7$$\"++++!R%F\\]yF^^[m7$Fa^[mF^[lFc\\l-Fh[l6$7&Fc^[m7$
Fa^[m$\"++++5KF,7$$\"++++?WF\\]yFh^[m7$F[_[mF^[lFc\\l-Fh[l6$7&F]_[m7$F
[_[m$\"++++DKF,7$$\"++++]WF\\]yFb_[m7$Fe_[mF^[lFc\\l-Fh[l6$7&Fg_[m7$Fe
_[m$\"++++SKF,7$$\"++++![%F\\]yF\\`[m7$F_`[mF^[lFc\\l-Fh[l6$7&Fa`[m7$F
_`[m$\"++++bKF,7$$\"++++5XF\\]yFf`[m7$Fi`[mF^[lFc\\l-Fh[l6$7&F[a[m7$Fi
`[m$\"++++qKF,7$$\"++++SXF\\]yF`a[m7$Fca[mF^[lFc\\l-Fh[l6$7&Fea[m7$Fca
[m$\"++++&G$F,7$$\"++++qXF\\]yFja[m7$F]b[mF^[lFc\\l-Fh[l6$7&F_b[m7$F]b
[m$Ff[dlF,7$$F]^flF\\]yFdb[m7$Ffb[mF^[lFc\\l-Fh[l6$7&Fgb[m7$Ffb[m$\"++
++:LF,7$$\"++++IYF\\]yF\\c[m7$F_c[mF^[lFc\\l-Fh[l6$7&Fac[m7$F_c[m$\"++
++ILF,7$$\"++++gYF\\]yFfc[m7$Fic[mF^[lFc\\l-Fh[l6$7&F[d[m7$Fic[m$\"+++
+XLF,7$$\"++++!p%F\\]yF`d[m7$Fcd[mF^[lFc\\l-Fh[l6$7&Fed[m7$Fcd[m$\"+++
+gLF,7$$\"++++?ZF\\]yFjd[m7$F]e[mF^[lFc\\l-Fh[l6$7&F_e[m7$F]e[m$\"++++
vLF,7$$\"++++]ZF\\]yFde[m7$Fge[mF^[lFc\\l-Fh[l6$7&Fie[m7$Fge[m$\"++++!
R$F,7$$\"++++!y%F\\]yF^f[m7$Faf[mF^[lFc\\l-Fh[l6$7&Fcf[m7$Faf[m$\"++++
0MF,7$$\"++++5[F\\]yFhf[m7$F[g[mF^[lFc\\l-Fh[l6$7&F]g[m7$F[g[m$\"++++?
MF,7$$\"++++S[F\\]yFbg[m7$Feg[mF^[lFc\\l-Fh[l6$7&Fgg[m7$Feg[m$\"++++NM
F,7$$\"++++q[F\\]yF\\h[m7$F_h[mF^[lFc\\l-Fh[l6$7&Fah[m7$F_h[m$F_\\dlF,
7$$Fg^flF\\]yFfh[m7$Fhh[mF^[lFc\\l-Fh[l6$7&Fih[m7$Fhh[m$\"++++lMF,7$$
\"++++I\\F\\]yF^i[m7$Fai[mF^[lFc\\l-Fh[l6$7&Fci[m7$Fai[m$\"++++![$F,7$
$\"++++g\\F\\]yFhi[m7$F[j[mF^[lFc\\l-Fh[l6$7&F]j[m7$F[j[m$\"++++&\\$F,
7$$\"++++!*\\F\\]yFbj[m7$Fej[mF^[lFc\\l-Fh[l6$7&Fgj[m7$Fej[m$\"++++5NF
,7$$\"++++?]F\\]yF\\[\\m7$F_[\\mF^[lFc\\l-Fh[l6$7&Fa[\\m7$F_[\\m$\"+++
+DNF,7$$\"++++]]F\\]yFf[\\m7$Fi[\\mF^[lFc\\l-Fh[l6$7&F[\\\\m7$Fi[\\m$
\"++++SNF,7$$\"++++!3&F\\]yF`\\\\m7$Fc\\\\mF^[lFc\\l-Fh[l6$7&Fe\\\\m7$
Fc\\\\m$\"++++bNF,7$$\"++++5^F\\]yFj\\\\m7$F]]\\mF^[lFc\\l-Fh[l6$7&F_]
\\m7$F]]\\m$\"++++qNF,7$$\"++++S^F\\]yFd]\\m7$Fg]\\mF^[lFc\\l-Fh[l6$7&
Fi]\\m7$Fg]\\m$\"++++&e$F,7$$\"++++q^F\\]yF^^\\m7$Fa^\\mF^[lFc\\l-Fh[l
6$7&Fc^\\m7$Fa^\\m$Fh\\dlF,7$$Fa_flF\\]yFh^\\m7$Fj^\\mF^[lFc\\l-Fh[l6$
7&F[_\\m7$Fj^\\m$\"++++:OF,7$$\"++++I_F\\]yF`_\\m7$Fc_\\mF^[lFc\\l-Fh[
l6$7&Fe_\\m7$Fc_\\m$\"++++IOF,7$$\"++++g_F\\]yFj_\\m7$F]`\\mF^[lFc\\l-
Fh[l6$7&F_`\\m7$F]`\\m$\"++++XOF,7$$\"++++!H&F\\]yFd`\\m7$Fg`\\mF^[lFc
\\l-Fh[l6$7&Fi`\\m7$Fg`\\m$\"++++gOF,7$$\"++++?`F\\]yF^a\\m7$Faa\\mF^[
lFc\\l-Fh[l6$7&Fca\\m7$Faa\\m$\"++++vOF,7$$\"++++]`F\\]yFha\\m7$F[b\\m
F^[lFc\\l-Fh[l6$7&F]b\\m7$F[b\\m$\"++++!p$F,7$$\"++++!Q&F\\]yFbb\\m7$F
eb\\mF^[lFc\\l-Fh[l6$7&Fgb\\m7$Feb\\m$\"++++0PF,7$$\"++++5aF\\]yF\\c\\
m7$F_c\\mF^[lFc\\l-Fh[l6$7&Fac\\m7$F_c\\m$\"++++?PF,7$$\"++++SaF\\]yFf
c\\m7$Fic\\mF^[lFc\\l-Fh[l6$7&F[d\\m7$Fic\\m$\"++++NPF,7$$\"++++qaF\\]
yF`d\\m7$Fcd\\mF^[lFc\\l-Fh[l6$7&Fed\\m7$Fcd\\m$Fa]dlF,7$$F[`flF\\]yFj
d\\m7$F\\e\\mF^[lFc\\l-Fh[l6$7&F]e\\m7$F\\e\\m$\"++++lPF,7$$\"++++IbF
\\]yFbe\\m7$Fee\\mF^[lFc\\l-Fh[l6$7&Fge\\m7$Fee\\m$\"++++!y$F,7$$\"+++
+gbF\\]yF\\f\\m7$F_f\\mF^[lFc\\l-Fh[l6$7&Faf\\m7$F_f\\m$\"++++&z$F,7$$
\"++++!f&F\\]yFff\\m7$Fif\\mF^[lFc\\l-Fh[l6$7&F[g\\m7$Fif\\m$\"++++5QF
,7$$\"++++?cF\\]yF`g\\m7$Fcg\\mF^[lFc\\l-Fh[l6$7&Feg\\m7$Fcg\\m$\"++++
DQF,7$$\"++++]cF\\]yFjg\\m7$F]h\\mF^[lFc\\l-Fh[l6$7&F_h\\m7$F]h\\m$\"+
+++SQF,7$$\"++++!o&F\\]yFdh\\m7$Fgh\\mF^[lFc\\l-Fh[l6$7&Fih\\m7$Fgh\\m
$\"++++bQF,7$$\"++++5dF\\]yF^i\\m7$Fai\\mF^[lFc\\l-Fh[l6$7&Fci\\m7$Fai
\\m$\"++++qQF,7$$\"++++SdF\\]yFhi\\m7$F[j\\mF^[lFc\\l-Fh[l6$7&F]j\\m7$
F[j\\m$\"++++&)QF,7$$\"++++qdF\\]yFbj\\m7$Fej\\mF^[lFc\\l-Fh[l6$7&Fgj
\\m7$Fej\\m$Fj]dlF,7$$Fe`flF\\]yF\\[]m7$F^[]mF^[lFc\\l-Fh[l6$7&F_[]m7$
F^[]m$\"++++:RF,7$$\"++++IeF\\]yFd[]m7$Fg[]mF^[lFc\\l-Fh[l6$7&Fi[]m7$F
g[]m$\"++++IRF,7$$\"++++geF\\]yF^\\]m7$Fa\\]mF^[lFc\\l-Fh[l6$7&Fc\\]m7
$Fa\\]m$\"++++XRF,7$$\"++++!*eF\\]yFh\\]m7$F[]]mF^[lFc\\l-Fh[l6$7&F]]]
m7$F[]]m$\"++++gRF,7$$\"++++?fF\\]yFb]]m7$Fe]]mF^[lFc\\l-Fh[l6$7&Fg]]m
7$Fe]]m$\"++++vRF,7$$\"++++]fF\\]yF\\^]m7$F_^]mF^[lFc\\l-Fh[l6$7&Fa^]m
7$F_^]m$\"++++!*RF,7$$\"++++!)fF\\]yFf^]m7$Fi^]mF^[lFc\\l-Fh[l6$7&F[_]
m7$Fi^]m$\"++++0SF,7$$\"++++5gF\\]yF`_]m7$Fc_]mF^[lFc\\l-Fh[l6$7&Fe_]m
7$Fc_]m$\"++++?SF,7$$\"++++SgF\\]yFj_]m7$F]`]mF^[lFc\\l-Fh[l6$7&F_`]m7
$F]`]m$\"++++NSF,7$$\"++++qgF\\]yFd`]m7$Fg`]mF^[lFc\\l-Fh[l6$7&Fi`]m7$
Fg`]m$Fc^dlF,7$$F_aflF\\]yF^a]m7$F`a]mF^[lFc\\l-Fh[l6$7&Faa]m7$F`a]m$
\"++++lSF,7$$\"++++IhF\\]yFfa]m7$Fia]mF^[lFc\\l-Fh[l6$7&F[b]m7$Fia]m$
\"++++!3%F,7$$\"++++ghF\\]yF`b]m7$Fcb]mF^[lFc\\l-Fh[l6$7&Feb]m7$Fcb]m$
\"++++&4%F,7$$\"++++!>'F\\]yFjb]m7$F]c]mF^[lFc\\l-Fh[l6$7&F_c]m7$F]c]m
$\"++++5TF,7$$\"++++?iF\\]yFdc]m7$Fgc]mF^[lFc\\l-Fh[l6$7&Fic]m7$Fgc]m$
\"++++DTF,7$$\"++++]iF\\]yF^d]m7$Fad]mF^[lFc\\l-Fh[l6$7&Fcd]m7$Fad]m$
\"++++STF,7$$\"++++!G'F\\]yFhd]m7$F[e]mF^[lFc\\l-Fh[l6$7&F]e]m7$F[e]m$
\"++++bTF,7$$\"++++5jF\\]yFbe]m7$Fee]mF^[lFc\\l-Fh[l6$7&Fge]m7$Fee]m$
\"++++qTF,7$$\"++++SjF\\]yF\\f]m7$F_f]mF^[lFc\\l-Fh[l6$7&Faf]m7$F_f]m$
\"++++&=%F,7$$\"++++qjF\\]yFff]m7$Fif]mF^[lFc\\l-Fh[l6$7&F[g]m7$Fif]m$
F\\_dlF,7$$FiaflF\\]yF`g]m7$Fbg]mF^[lFc\\l-Fh[l6$7&Fcg]m7$Fbg]m$\"++++
:UF,7$$\"++++IkF\\]yFhg]m7$F[h]mF^[lFc\\l-Fh[l6$7&F]h]m7$F[h]m$\"++++I
UF,7$$\"++++gkF\\]yFbh]m7$Feh]mF^[lFc\\l-Fh[l6$7&Fgh]m7$Feh]m$\"++++XU
F,7$$\"++++!\\'F\\]yF\\i]m7$F_i]mF^[lFc\\l-Fh[l6$7&Fai]m7$F_i]m$\"++++
gUF,7$$\"++++?lF\\]yFfi]m7$Fii]mF^[lFc\\l-Fh[l6$7&F[j]m7$Fii]m$\"++++v
UF,7$$\"++++]lF\\]yF`j]m7$Fcj]mF^[lFc\\l-Fh[l6$7&Fej]m7$Fcj]m$\"++++!H
%F,7$$\"++++!e'F\\]yFjj]m7$F][^mF^[lFc\\l-Fh[l6$7&F_[^m7$F][^m$\"++++0
VF,7$$\"++++5mF\\]yFd[^m7$Fg[^mF^[lFc\\l-Fh[l6$7&Fi[^m7$Fg[^m$\"++++?V
F,7$$\"++++SmF\\]yF^\\^m7$Fa\\^mF^[lFc\\l-Fh[l6$7&Fc\\^m7$Fa\\^m$\"+++
+NVF,7$$\"++++qmF\\]yFh\\^m7$F[]^mF^[lFc\\l-Fh[l6$7&F]]^m7$F[]^m$Fe_dl
F,7$$FcbflF\\]yFb]^m7$Fd]^mF^[lFc\\l-Fh[l6$7&Fe]^m7$Fd]^m$\"++++lVF,7$
$\"++++InF\\]yFj]^m7$F]^^mF^[lFc\\l-Fh[l6$7&F_^^m7$F]^^m$\"++++!Q%F,7$
$\"++++gnF\\]yFd^^m7$Fg^^mF^[lFc\\l-Fh[l6$7&Fi^^m7$Fg^^m$\"++++&R%F,7$
$\"++++!z'F\\]yF^_^m7$Fa_^mF^[lFc\\l-Fh[l6$7&Fc_^m7$Fa_^m$\"++++5WF,7$
$\"++++?oF\\]yFh_^m7$F[`^mF^[lFc\\l-Fh[l6$7&F]`^m7$F[`^m$\"++++DWF,7$$
\"++++]oF\\]yFb`^m7$Fe`^mF^[lFc\\l-Fh[l6$7&Fg`^m7$Fe`^m$\"++++SWF,7$$
\"++++!)oF\\]yF\\a^m7$F_a^mF^[lFc\\l-Fh[l6$7&Faa^m7$F_a^m$\"++++bWF,7$
$\"++++5pF\\]yFfa^m7$Fia^mF^[lFc\\l-Fh[l6$7&F[b^m7$Fia^m$\"++++qWF,7$$
\"++++SpF\\]yF`b^m7$Fcb^mF^[lFc\\l-Fh[l6$7&Feb^m7$Fcb^m$\"++++&[%F,7$$
\"++++qpF\\]yFjb^m7$F]c^mF^[lFc\\l-Fh[l6$7&F_c^m7$F]c^m$F]aclF,7$$F_fe
lF\\]yFdc^m7$Ffc^mF^[lFc\\l-Fh[l6$7&Fgc^m7$Ffc^m$\"++++:XF,7$$\"++++Iq
F\\]yF\\d^m7$F_d^mF^[lFc\\l-Fh[l6$7&Fad^m7$F_d^m$\"++++IXF,7$$\"++++gq
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01" "Curve 502" "Curve 503" "Curve 504" "Curve 505" "Curve 506" "Curve
 507" "Curve 508" "Curve 509" "Curve 510" "Curve 511" "Curve 512" "Cur
ve 513" "Curve 514" "Curve 515" "Curve 516" "Curve 517" "Curve 518" "C
urve 519" "Curve 520" "Curve 521" "Curve 522" "Curve 523" "Curve 524" 
"Curve 525" "Curve 526" "Curve 527" "Curve 528" "Curve 529" "Curve 530
" "Curve 531" "Curve 532" "Curve 533" "Curve 534" "Curve 535" "Curve 5
36" "Curve 537" "Curve 538" "Curve 539" "Curve 540" "Curve 541" "Curve
 542" "Curve 543" "Curve 544" "Curve 545" "Curve 546" "Curve 547" "Cur
ve 548" "Curve 549" "Curve 550" "Curve 551" "Curve 552" "Curve 553" "C
urve 554" "Curve 555" "Curve 556" "Curve 557" "Curve 558" "Curve 559" 
"Curve 560" "Curve 561" "Curve 562" "Curve 563" "Curve 564" "Curve 565
" "Curve 566" "Curve 567" "Curve 568" "Curve 569" "Curve 570" "Curve 5
71" "Curve 572" "Curve 573" "Curve 574" "Curve 575" "Curve 576" "Curve
 577" "Curve 578" "Curve 579" "Curve 580" "Curve 581" "Curve 582" "Cur
ve 583" "Curve 584" "Curve 585" "Curve 586" "Curve 587" "Curve 588" "C
urve 589" "Curve 590" "Curve 591" "Curve 592" "Curve 593" "Curve 594" 
"Curve 595" "Curve 596" "Curve 597" "Curve 598" "Curve 599" "Curve 600
" "Curve 601" "Curve 602" "Curve 603" "Curve 604" "Curve 605" "Curve 6
06" "Curve 607" "Curve 608" "Curve 609" "Curve 610" "Curve 611" "Curve
 612" "Curve 613" "Curve 614" "Curve 615" "Curve 616" "Curve 617" "Cur
ve 618" "Curve 619" "Curve 620" "Curve 621" "Curve 622" "Curve 623" "C
urve 624" "Curve 625" "Curve 626" "Curve 627" "Curve 628" "Curve 629" 
"Curve 630" "Curve 631" "Curve 632" "Curve 633" "Curve 634" "Curve 635
" "Curve 636" "Curve 637" "Curve 638" "Curve 639" "Curve 640" "Curve 6
41" "Curve 642" "Curve 643" "Curve 644" "Curve 645" "Curve 646" "Curve
 647" "Curve 648" "Curve 649" "Curve 650" "Curve 651" "Curve 652" "Cur
ve 653" "Curve 654" "Curve 655" "Curve 656" "Curve 657" "Curve 658" "C
urve 659" "Curve 660" "Curve 661" "Curve 662" "Curve 663" "Curve 664" 
"Curve 665" "Curve 666" "Curve 667" "Curve 668" "Curve 669" "Curve 670
" "Curve 671" "Curve 672" "Curve 673" "Curve 674" "Curve 675" "Curve 6
76" "Curve 677" "Curve 678" "Curve 679" "Curve 680" "Curve 681" "Curve
 682" "Curve 683" "Curve 684" "Curve 685" "Curve 686" "Curve 687" "Cur
ve 688" "Curve 689" "Curve 690" "Curve 691" "Curve 692" "Curve 693" "C
urve 694" "Curve 695" "Curve 696" "Curve 697" "Curve 698" "Curve 699" 
"Curve 700" "Curve 701" "Curve 702" "Curve 703" "Curve 704" "Curve 705
" "Curve 706" "Curve 707" "Curve 708" "Curve 709" "Curve 710" "Curve 7
11" "Curve 712" "Curve 713" "Curve 714" "Curve 715" "Curve 716" "Curve
 717" "Curve 718" "Curve 719" "Curve 720" "Curve 721" "Curve 722" "Cur
ve 723" "Curve 724" "Curve 725" "Curve 726" "Curve 727" "Curve 728" "C
urve 729" "Curve 730" "Curve 731" "Curve 732" "Curve 733" "Curve 734" 
"Curve 735" "Curve 736" "Curve 737" "Curve 738" "Curve 739" "Curve 740
" "Curve 741" "Curve 742" "Curve 743" "Curve 744" "Curve 745" "Curve 7
46" "Curve 747" "Curve 748" "Curve 749" "Curve 750" "Curve 751" "Curve
 752" "Curve 753" "Curve 754" "Curve 755" "Curve 756" "Curve 757" "Cur
ve 758" "Curve 759" "Curve 760" "Curve 761" "Curve 762" "Curve 763" "C
urve 764" "Curve 765" "Curve 766" "Curve 767" "Curve 768" "Curve 769" 
"Curve 770" "Curve 771" "Curve 772" "Curve 773" "Curve 774" "Curve 775
" "Curve 776" "Curve 777" "Curve 778" "Curve 779" "Curve 780" "Curve 7
81" "Curve 782" "Curve 783" "Curve 784" "Curve 785" "Curve 786" "Curve
 787" "Curve 788" "Curve 789" "Curve 790" "Curve 791" "Curve 792" "Cur
ve 793" "Curve 794" "Curve 795" "Curve 796" "Curve 797" "Curve 798" "C
urve 799" "Curve 800" "Curve 801" "Curve 802" "Curve 803" "Curve 804" 
"Curve 805" "Curve 806" "Curve 807" "Curve 808" "Curve 809" "Curve 810
" "Curve 811" "Curve 812" "Curve 813" "Curve 814" "Curve 815" "Curve 8
16" "Curve 817" "Curve 818" "Curve 819" "Curve 820" "Curve 821" "Curve
 822" "Curve 823" "Curve 824" "Curve 825" "Curve 826" "Curve 827" "Cur
ve 828" "Curve 829" "Curve 830" "Curve 831" "Curve 832" "Curve 833" "C
urve 834" "Curve 835" "Curve 836" "Curve 837" "Curve 838" "Curve 839" 
"Curve 840" "Curve 841" "Curve 842" "Curve 843" "Curve 844" "Curve 845
" "Curve 846" "Curve 847" "Curve 848" "Curve 849" "Curve 850" "Curve 8
51" "Curve 852" "Curve 853" "Curve 854" "Curve 855" "Curve 856" "Curve
 857" "Curve 858" "Curve 859" "Curve 860" "Curve 861" "Curve 862" "Cur
ve 863" "Curve 864" "Curve 865" "Curve 866" "Curve 867" "Curve 868" "C
urve 869" "Curve 870" "Curve 871" "Curve 872" "Curve 873" "Curve 874" 
"Curve 875" "Curve 876" "Curve 877" "Curve 878" "Curve 879" "Curve 880
" "Curve 881" "Curve 882" "Curve 883" "Curve 884" "Curve 885" "Curve 8
86" "Curve 887" "Curve 888" "Curve 889" "Curve 890" "Curve 891" "Curve
 892" "Curve 893" "Curve 894" "Curve 895" "Curve 896" "Curve 897" "Cur
ve 898" "Curve 899" "Curve 900" "Curve 901" "Curve 902" "Curve 903" "C
urve 904" "Curve 905" "Curve 906" "Curve 907" "Curve 908" "Curve 909" 
"Curve 910" "Curve 911" "Curve 912" "Curve 913" "Curve 914" "Curve 915
" "Curve 916" "Curve 917" "Curve 918" "Curve 919" "Curve 920" "Curve 9
21" "Curve 922" "Curve 923" "Curve 924" "Curve 925" "Curve 926" "Curve
 927" "Curve 928" "Curve 929" "Curve 930" "Curve 931" "Curve 932" "Cur
ve 933" "Curve 934" "Curve 935" "Curve 936" "Curve 937" "Curve 938" "C
urve 939" "Curve 940" "Curve 941" "Curve 942" "Curve 943" "Curve 944" 
"Curve 945" "Curve 946" "Curve 947" "Curve 948" "Curve 949" "Curve 950
" "Curve 951" "Curve 952" "Curve 953" "Curve 954" "Curve 955" "Curve 9
56" "Curve 957" "Curve 958" "Curve 959" "Curve 960" "Curve 961" "Curve
 962" "Curve 963" "Curve 964" "Curve 965" "Curve 966" "Curve 967" "Cur
ve 968" "Curve 969" "Curve 970" "Curve 971" "Curve 972" "Curve 973" "C
urve 974" "Curve 975" "Curve 976" "Curve 977" "Curve 978" "Curve 979" 
"Curve 980" "Curve 981" "Curve 982" "Curve 983" "Curve 984" "Curve 985
" "Curve 986" "Curve 987" "Curve 988" "Curve 989" "Curve 990" "Curve 9
91" "Curve 992" "Curve 993" "Curve 994" "Curve 995" "Curve 996" "Curve
 997" "Curve 998" "Curve 999" "Curve 1000" "Curve 1001" }}}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"%+5\"\"\"-%$SumG6$,&\"\"*!\"\"*&#F&
\"$+#F(-%\"iG6#,&\"\"#F.*(F&F(F'F.F3F(F(F(F(/F3;F(F'F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$!+++]xW!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "This seems to support the observation." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 179 "If g is
 continuous (or monotone) on [a,b] then this approximating rectangle s
um process will have a limit which we denote as the definite integral \+
of g over [a,b] the notation is " }{XPPEDIT 257 0 "Int(g,x = a .. b);
" "6#-%$IntG6$%\"gG/%\"xG;%\"aG%\"bG" }{TEXT 258 182 " and this defini
te integral can be interpreted as the difference of areas where g is a
bove the x axis and below the x axis. We say that g is an integrable f
unction when this happens." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 29 "We try this for our function " }{XPPEDIT 18 0 "f \+
(x)=1+5*x" "6#/-%\"fG6#%\"xG,&\"\"\"F)*&\"\"&F)F'F)F)" }{TEXT -1 1 " \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(f,x=-2..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"*\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 186 "This is the answer -4.5 that we were expecting. We also \+
use the term ...\"integrating f from -2 to 1\" in describing this proc
ess of evaluating the definite integral. We can now define an " }
{TEXT 259 19 "indefinite integral" }{TEXT 260 0 "" }{TEXT -1 96 " whic
h is a new function, based on this process of integrating continuous f
unctions g, such as :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " G:=x-> Int( g(t),t=-2..x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(-%$IntG6$-%\"gG6#%\"tG/F2;!\"#9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 124 " This is defining a new function G for each x, which say
s to find its value you integrate g over the interval from -2 to x. " 
}{TEXT 261 124 "The interesting fact is that this G is itself an antid
erivative for g. That is G is differentiable and its derivative is g. \+
" }{TEXT 262 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "This \+
is one part of the Fundamental theorem of Calculus. This yields a valu
able tool in evaluating these definite integrals. " }}{PARA 0 "" 0 "" 
{TEXT -1 39 " This says that over [a,b]  G(b)-G(a) =" }{XPPEDIT 256 0 
"Int(g,x = a .. b)" "6#-%$IntG6$%\"gG/%\"xG;%\"aG%\"bG" }{TEXT -1 1 " \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 321 " This equation says that to \+
find the definite integral, first we identify an antiderivative of g o
ver [a, b] then simply evaluate that antiderivative at the two endpoin
ts and subtract. Because of this theorem we can identify the process o
f finding antiderivatives and finding indefinite integrals. We check t
his with our " }{XPPEDIT 18 0 "f (x)=1+5*x" "6#/-%\"fG6#%\"xG,&\"\"\"F
)*&\"\"&F)F'F)F)" }{TEXT -1 36 " we know an antiderivative would be " 
}{XPPEDIT 18 0 "x+5*x^2/2;" "6#,&%\"xG\"\"\"*(\"\"&F%*$F$\"\"#F%F)!\"
\"F%" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F:
=int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&%\"xG\"\"\"*&#\"
\"&\"\"#F'*$)F&F+F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 " You s
hould note that here we have used the integral command " }{TEXT 263 8 
"int(f,x)" }{TEXT -1 89 " without specifying the interval to get this \+
antiderivative and we also gave it a name F." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(x=-2,F); sub
s(x=1,F);subs(x=1,F)-subs(x=-2,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"(\"\"#" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6##!\"*\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "
Here we see that we get the same answer as we expected -4.5. in our pr
evious worksheet we found the area under the curve " }{XPPEDIT 18 0 "x
^2-4*x+5" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"\"\"&F'" }{TEXT -1 
97 " over the interval [0,3] to be 6. We check the process by integrat
ing this expression over [0,3]." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(x^2-4*x+5,x=0..3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 105 "We further check it by finding an antiderivative for the
 expression and applying the fundamental theorem." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 70 "F:=int(x^2-4*x+5,x); subs(x=0,F); subs(x=3,F
);subs(x=3,F)-subs(x=0,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,(
*&#\"\"\"\"\"$F(*$)%\"xGF)F(F(F(*&\"\"#F()F,F.F(!\"\"*&\"\"&F(F,F(F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "
This shows the antiderivative and its values at 0 and 3 and then subst
racts giving the same answer of 6. " }}}}{MARK "22 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }