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{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "To demonstrate the Fundamental The
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0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&)e(=Z\"!\")" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Finally, we evaluate the integral
 directly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int(f(x),x=0.
.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$)%\"xG\"\"#\"\"\"
\"\"\"-%$sinG6#F)!\"%/F);\"\"!\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&)e(=Z\"!\")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 112 "As with the solve( ) command we can ente
r the function directly into the Int( ) command.  We use t instead of \+
x." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "Int(3*sqrt(t)+sin(t),t=0..Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,&*$-%%sqrtG6#%\"tG\"\"\"\"\"$-%$sinG6#F+\"\"
\"/F+;\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*flOJ\"!\")" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 109 "Finally, there is the case where no antideriva
tive exists or it is expressed in arcane, mysterious functions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "Int(sqrt(1+x^(3/2)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In
tG6$*$-%%sqrtG6#,&\"\"\"F+*$)%\"xG#\"\"$\"\"#\"\"\"F+F2F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%\"xG\"\"\"-%*hypergeomG6%7$#!\"\"\"\"##F,\"\"$7##\"
\"&F.,$*$)F$#F.F,\"\"\"F+F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "g:=x->exp(cos(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"
xG6\"6$%)operatorG%&arrowGF(-%$expG6#-%$cosG6#9$F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Int(g(x),x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-%$expG6#-%$cosG6#%\"xGF," }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$intG6$-%$expG6#-%$cosG6#%\"xGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 91 "There is no antiderivative known to Maple.  But it easily evalu
ates the definite integrals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "Int(exp(cos(x)),x=-1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$
IntG6$-%$expG6#-%$cosG6#%\"xG/F,;!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wY
xmi!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(sqrt(1+x^(3/
2)),x=0.5..1.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrt
G6#,&\"\"\"F+*$)%\"xG#\"\"$\"\"#\"\"\"F+F2/F.;$\"\"&!\"\"$\"#<F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+)QM]w\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
102 "The with(student) package has some techniques of integration rout
ines (changevar, intparts), and there" }}{PARA 0 "" 0 "" {TEXT -1 105 
" is also  the convert/parfrac combination.  Students may wish to lear
n them on their own to verify their " }}{PARA 0 "" 0 "" {TEXT -1 87 "h
omework answers.  Also there are some numerical quadrature schemes (mi
dpoint,simpson, " }}{PARA 0 "" 0 "" {TEXT -1 101 "trapezoid) which sho
uld be implemented on a calculator,  since it is a little absurd to be
 doing them" }}{PARA 0 "" 0 "" {TEXT -1 62 " with Maple when it has it
s own very accurate internal scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 8 "PROBLEMS" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 103 "1.    For the functions below, find \+
their indefinite integral and its antiderivative then evaluate the " }
}{PARA 0 "" 0 "" {TEXT -1 49 "        definite integral for the interv
al given." }}{PARA 0 "" 0 "" {TEXT -1 94 "            a)  f(x)=2*x^(-3
)-5*x^(3/2),  x=1..2    b)  g(t)=4*t^3-sin(2*t)+exp(-4*t),  t=1..9" }}
{PARA 0 "" 0 "" {TEXT -1 49 "            c)  h(u)=exp(-u)*cos(4*u),  u
=0..Pi/2" }}{PARA 0 "" 0 "" {TEXT -1 57 "            d)  f(x)=(x^4-3*x
+1)/(x^3-2*x^2-x+2),  x=3..5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 98 "2.    The function sqrt(1+x^3) has no antideriv
ative.  Graph a midpoint Riemann sum approximation " }}{PARA 0 "" 0 "
" {TEXT -1 90 "        and compute its sum for x=0..1 and n=10, 40.  T
hen evaluate the definite integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 90 "3.    Using the command plot(f,x=0..3,di
scont=true) plot the piecewise continuous function" }}{PARA 0 "" 0 "" 
{TEXT -1 56 "             f:=piecewise(x<=1,x^2,x<2,2*x+1,2<=x,-2*x).
" }}{PARA 0 "" 0 "" {TEXT -1 43 "        Then compute its definite int
egral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 
"4.    Some unbounded functions can be integrated.  Graph a midpoint R
iemann sum approximation" }}{PARA 0 "" 0 "" {TEXT -1 99 "        for x
=0..1 and n=20, 50, 100 for the function 1/sqrt(x) and compute its sum
.  Then find its" }}{PARA 0 "" 0 "" {TEXT -1 87 "        indefinite in
tegral and its antiderivative then evaluate the definite integral." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "47 17 0" 78 }
{VIEWOPTS 1 1 0 1 1 1803 }