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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "Integration in Maple" }
{TEXT -1 27 "\nLast revision: Spring 2008" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 15 "Antiderivatives" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The key command for integration is
 " }{TEXT 257 3 "int" }{TEXT -1 99 ".  Let us start by looking at inde
finite integrals.  Suppose we want to find an anti-derivative of " }
{XPPEDIT 18 0 "f(x) = x*exp(-x);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%$expG6#
,$F'!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We can do this eas
ily, just enter" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "int(x*exp(-x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 18 "The syntax of the " }{TEXT 258 4 "int " }{TEXT -1 12 "command i
s  " }{TEXT 259 28 " int(expression, variable). " }{TEXT -1 48 " When \+
written with an upper case I, the command " }{TEXT 260 3 "Int" }{TEXT 
-1 25 " is the inert version of " }{TEXT 261 4 "int," }{TEXT -1 72 " m
eaning that it displays the integral, but does not compute anything:  \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int(x*exp(-x),x);" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT 262 6 "It is " }{TEXT -1 11 "recommende
d" }{TEXT 272 8 " to use " }{TEXT -1 3 "Int" }{TEXT 264 12 " along wit
h " }{TEXT -1 5 "value" }{TEXT 265 1 "." }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(x*exp(-x),x);  value(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 263 4 "Int " }{TEXT -1 35 "displays the i
ntegral question and " }{TEXT 266 8 "value(%)" }{TEXT -1 23 " produces
 the solution." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
91 "Maple can't find every integral;  if it gets too hard, maple will \+
just repeat the question." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(ex
p(-x)*sin(sqrt(1+x^2)),x); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 42 "This is maple's way to say \"I don't know\"." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Constant \+
of Integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 
"" {TEXT 273 41 "Maple omits the constant of integration C" }{TEXT -1 
98 ".  This may occasionally lead to seemingly contradictory results. \+
 Let's say we want to integrate " }{XPPEDIT 18 0 "y = (x-1)^2;" "6#/%
\"yG*$,&%\"xG\"\"\"F(!\"\"\"\"#" }{TEXT -1 31 "  .  Direct computation
 yields " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "y := (x-1)^2; I
nt(y,x); Y := value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "But wh
en we expand y first, we obtain a different answer." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 41 "z := expand(y);  Int(z,x); Z := value(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Z-Y;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 174 "We see that the two answers differ by the constant 1/3.  This \+
confirms that anti-derivatives differ by a constant, and it also teach
es us to be attentive in the use of maple." }{MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "
 The Variable of Integration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }{TEXT -1 36 "Suppose that we want to intergate   " }{XPPEDIT 18 
0 "a^x;" "6#)%\"aG%\"xG" }{TEXT -1 80 " .  When x is the variable, we \+
have an exponential function with anti-derivative" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "Int(a^x,x); value(%);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 86 "If on the other hand a is the variable, we have a po
wer function with anti-derivative " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "Int(a^x,a);  value(%);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 91 "If the variable of integration is different from a and x,
 say it is t, then the expression " }{XPPEDIT 18 0 "a^x;" "6#)%\"aG%\"
xG" }{TEXT -1 40 "  is a constant and the integral becomes" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(a^x,t); value(%);" }}}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$)%\"aG%\"xG%\"tG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 269 18 
"Definite Integrals" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "For defini
te integrals we just include the limits of integration in the range of
 the variable.  Example:  Integrate  " }{XPPEDIT 18 0 "x*exp(-x);" "6#
*&%\"xG\"\"\"-%$expG6#,$F$!\"\"F%" }{TEXT -1 14 "  from 0 to 4." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(x*exp(-x),x=0..4);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We just get the result.  Again, it
 is preferable to use the Int & value combination:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "Int(x*exp(-x),x=0..4); value(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 26 "For numerical results use " }{TEXT 267 6 "evalf
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(x*exp(-x),x=0..4); evalf(%
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Maple can't solve every in
tegral;  if it gets too hard, maple will just repeat the question, as \+
we saw above.  But we can always ask for a numerical approximation." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Int(exp(-x)*sin(sqrt(1+x^2)),x=0..
3); value(%);\nevalf(Int(exp(-x)*sin(sqrt(1+x^2)),x=0..3));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 30 "Variable Limits of Integration" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "We may also define functions with variable limits of inte
gration.  " }}{PARA 0 "" 0 "" {TEXT 274 8 "Example:" }{TEXT -1 2 "  " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "S := x -> int(sin(Pi*t^2/
2),t=0..x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "This function is k
nown in the literature as the Fresnel Sine function.  Maple recognizes
 that:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S(x);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 46 "Let's take the derivative to check the answer." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(S)(x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 275 9 "Example: " }{TEXT -1 102 " (This is problem 81, page 259)
.  Define a function F(x) by an integral, and then take the derivative
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := x -> int( sqrt(t), t = 0.
.sin(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The derivative is" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(x);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 72 "We can also first compute F(x) explicitly, and then tak
e the derivative:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dF := diff(%,x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "We see that the results are identical." }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 270 21 "Nume
rical Integration" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 412 "Numerical in
tegration is the approximate calculation of the value of a definite in
tegral.  This is useful when the integrand is a complicated function w
ithout a simple anti-derivative.  Leftsums, rightsums, middlesums (Sec
tion 5.1) are examples of numerical integration; further techniques, s
uch as the trapezoidal rule and Simpson's rule are outlined in Section
 5.9.  All of these are built-in functions in the " }{TEXT 271 15 "stu
dent package" }{TEXT -1 59 ".  We illustrate all of these methods for \+
the integral of  " }{XPPEDIT 18 0 "x^4-7*x^2+8*x+1;" "6#,**$%\"xG\"\"%
\"\"\"*&\"\"(F'*$F%\"\"#F'!\"\"*&\"\")F'F%F'F'F'F'" }{TEXT -1 131 "  o
n the interval from 0 to 2 with 10 subintervals, and we compare the nu
merical approximations to the exact value of the integral." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart;\nwith(student);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y := x^4 - 7*x^2 + 8*x +1;  \+
# abbreviation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(y,x=0
..2); Itrue := evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 
"middlebox(y,x=0..2,10); middlesum(y,x=0..2,10); M := evalf(%); M-Itru
e; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "leftbox(y,x=0..2,10)
; leftsum(y,x=0..2,10); L := evalf(%); L-Itrue; " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 70 "rightbox(y,x=0..2,10); rightsum(y,x=0..2,10);
 R := evalf(%); R-Itrue; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The
 approximations with rightsums and leftsums are not as good as the res
ult from the middlesum, however their average is in the same ballpark:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ave := (L+R)/2;  ave-Itrue;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The trapezoidal rule yields" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "trapezoid(y,x=0..2,10); T := evalf(
%); T - Itrue;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Note that the \+
trapezoidal rule is the average of leftsums and rightsums.\nSimpson's \+
rule applied to our problem yields" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "simpson(y,x=0..2,10); S := evalf(%); S - Itrue;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 70 "Simpson's rule is superior and yields the best \+
result in this example." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
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