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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable
" }{TEXT 307 3 "  \n" }{TEXT 300 35 "Brian E. Blank and Steven G. Kran
tz" }{TEXT 301 39 "\n\nSection 5.3\nProperties of Integrals \n" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "1. Estimates for an Integral" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If  " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 59 "  is a continuous function \+
on the closed bounded interval  " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%
\"bG" }{TEXT -1 25 ", then, according to the " }{TEXT 308 21 "Extreme \+
Value Theorem" }{TEXT -1 21 ",  there are points  " }{XPPEDIT 18 0 "x[
m];" "6#&%\"xG6#%\"mG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "x[M];" "6
#&%\"xG6#%\"MG" }{TEXT -1 7 "   in  " }{XPPEDIT 18 0 "[a, b];" "6#7$%
\"aG%\"bG" }{TEXT -1 12 "   such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 34 "                                  " }
{XPPEDIT 18 0 "m <= f(x);" "6#1%\"mG-%\"fG6#%\"xG" }{TEXT -1 31 "     \+
           and            " }{XPPEDIT 18 0 "f(x) <= M;" "6#1-%\"fG6#%
\"xG%\"MG" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT 
-1 5 "where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 33 "                                 " }{XPPEDIT 18 0 "m = f(x[m]);
" "6#/%\"mG-%\"fG6#&%\"xG6#F$" }{TEXT -1 28 "              and        \+
   " }{XPPEDIT 18 0 "M = f(x[M]);" "6#/%\"MG-%\"fG6#&%\"xG6#F$" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "These extreme values of  \+
" }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 17 " on the interval " }
{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 36 " allow us to bo
und the integral of  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 8 "  ov
er  " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "         \+
           " }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }{XPPEDIT 
18 0 "m*(b-a) <= Int(f(x),x = a .. b);" "6#1*&%\"mG\"\"\",&%\"bGF&%\"a
G!\"\"F&-%$IntG6$-%\"fG6#%\"xG/F1;F)F(" }{TEXT -1 27 "             and
           " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) <= M*(b-a);" "6#1-%$
IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG*&%\"MG\"\"\",&F.F1F-!\"\"F1" }{TEXT 
-1 3 "  ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 34 "2. Example (Exercise 34, Page 375)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Estimate the integral    \+
" }{XPPEDIT 18 0 "Int(sqrt(1+3*cos(x)^5),x = 0 .. Pi/2);" "6#-%$IntG6$
-%%sqrtG6#,&\"\"\"F**&\"\"$F**$)-%$cosG6#%\"xG\"\"&F*F*F*/F2;\"\"!*&%#
PiGF*\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 309 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 48 "For the given integrand on the given \+
interval,  " }{XPPEDIT 18 0 "x[m] = Pi/2;" "6#/&%\"xG6#%\"mG*&%#PiG\"
\"\"\"\"#!\"\"" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "m = 1;" "6#/%\"mG\"
\"\"" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "M = 0;" "6#/%\"MG\"\"!" }
{TEXT -1 11 ",   and    " }{XPPEDIT 18 0 "M = 2;" "6#/%\"MG\"\"#" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Since  " }{XPPEDIT 18 0 
"b-a = Pi/2;" "6#/,&%\"bG\"\"\"%\"aG!\"\"*&%#PiGF&\"\"#F(" }{TEXT -1 
10 ",  we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "                    " }{XPPEDIT 18 0 "Pi/2 <= Int(sqrt(1+
3*cos(x)^5),x = 0 .. Pi/2);" "6#1*&%#PiG\"\"\"\"\"#!\"\"-%$IntG6$-%%sq
rtG6#,&F&F&*&\"\"$F&*$)-%$cosG6#%\"xG\"\"&F&F&F&/F7;\"\"!*&F%F&F'F(" }
{TEXT -1 33 "               and               " }{XPPEDIT 18 0 "Int(sq
rt(1+3*cos(x)^5),x = 0 .. Pi/2) <= Pi;" "6#1-%$IntG6$-%%sqrtG6#,&\"\"
\"F+*&\"\"$F+*$)-%$cosG6#%\"xG\"\"&F+F+F+/F3;\"\"!*&%#PiGF+\"\"#!\"\"F
9" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "          " }}{PARA 0 "" 0 "" {TEXT -1 47 "In fact, using
 Maple's numerical integrator,   " }{TEXT 310 9 "evalf@Int" }{TEXT -1 
14 ",    we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf(Int(sq
rt(1+3*cos(x)^5),x = 0 .. Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+jrqa@!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "3. The Mean Valu
e Theorem for Integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "If  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 30 "  i
s a continuous function on " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG
" }{TEXT -1 25 ",  then there is a point " }{XPPEDIT 18 0 "c;" "6#%\"c
G" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }
{TEXT -1 10 " such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 45 "                                             " }
{XPPEDIT 18 0 "f(c) = 1/(b-a)*Int(f(x),x = a .. b);" "6#/-%\"fG6#%\"cG
,$-%$IntG6$-F%6#%\"xG/F.;%\"aG%\"bG*&\"\"\"F4,&F2F4F1!\"\"F6" }}{PARA 
0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 19 "We may interp
ret   " }{XPPEDIT 18 0 "f(c);" "6#-%\"fG6#%\"cG" }{TEXT -1 25 "  as th
e average value   " }{XPPEDIT 18 0 "f[ave];" "6#&%\"fG6#%$aveG" }
{TEXT -1 9 "   of    " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 8 "   o
ver " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 75 
" is positive, this equation may be interpreted geometrically as follo
ws:  \n" }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 18 "  attains a value
 " }{XPPEDIT 18 0 "f(c);" "6#-%\"fG6#%\"cG" }{TEXT -1 7 " with  " }
{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 6 "  in  " }{XPPEDIT 18 0 "[a, \+
b];" "6#7$%\"aG%\"bG" }{TEXT -1 54 "  such that the area under the gra
ph of the equation  " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"x
G" }{TEXT -1 6 "  for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " in
  " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 52 "  is equal
 to the area of the rectangle with height " }{XPPEDIT 18 0 "f(c);" "6#
-%\"fG6#%\"cG" }{TEXT -1 17 " and base length " }{XPPEDIT 18 0 "b-a;" 
"6#,&%\"bG\"\"\"%\"aG!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "                               \+
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  f := x -> 2*x+2/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%$expG6#\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"
\"#\"\"\"9$F/F/*&F.F/F0!\"\"F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "fbar := int(f(x),x=a..b)/(b-a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%fbarG*&,&-%$expG6#\"\"#\"\"\"F+F+F+,&-F(6#F+F+F+!\"
\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "candidates := [solv
e( f(c) = fbar, c )];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+candidates
G7$,$*&#\"\"\"\"\"#F)*&,&*&F*F)-%$expG6#F)F)F)F*!\"\"F1,(-F/6#F*F)F)F)
*$,,*$)F3F*F)F)*&F*F)F3F)F)\"#:F1*&\"#;F))F.F*F)F1*&\"#KF)F.F)F)F(F)F)
F)F),$*&F(F)*&F,F1,(F3F)F)F)F5F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "map(evalf,candidates);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$$\"+%3!R?>!\"*$\"+q`F2_!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "plot([f(x),f(candidates[1])], x = 1.. exp(1), color =
 [RED,CORAL], scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 560 
374 374 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"33+++,+++5!#<$\"\"%\"\"!7$$
\"3BRc$[l`u.\"F*$\"3dH!3yE/F+%F*7$$\"3#*HWxy=/q5F*$\"3>f(H=[p\"4SF*7$$
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" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "5. A Numerical Example" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Calculate the a
verage value   " }{XPPEDIT 18 0 "f[ave];" "6#&%\"fG6#%$aveG" }{TEXT 
-1 4 "  of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 20 "                    " }{XPPEDIT 18 0 "f(x) = sqrt(1+3*cos(x)^5)
;" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"\"F,*&\"\"$F,*$)-%$cosG6#F'\"\"&F,
F,F," }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "o
ver the interval   " }{XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"
\"\"\"#!\"\"" }{TEXT -1 16 ".  Find a value " }{XPPEDIT 18 0 "c;" "6#%
\"cG" }{TEXT -1 14 " such that    " }{XPPEDIT 18 0 "f(c) = f[ave];" "6
#/-%\"fG6#%\"cG&F%6#%$aveG" }{TEXT -1 2 " ." }}{PARA 3 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "(This is the data form Exercise
 34, Page 375, that we examined in Section 2 above.)" }}{PARA 3 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 312 9 "Solution:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We calculate" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "a := 0;  b := Pi/2;  f := x -> sqrt(1+3*cos(
x)^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"bG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6
#,&\"\"\"F0*&\"\"$F0)-%$cosG6#9$\"\"&F0F0F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "fbar := evalf(Int(f(x),x=a..b)/(b-a));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fbarG$\"+$=H<P\"!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "c := fsolve(f(x) = fbar, x, 0 ..Pi/
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+v%>pr'!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot([f(x),f(c)], x = a.. b, color \+
= [RED,CORAL], scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information
" }}{PARA 0 "" 0 "" {TEXT -1 101 "\nWorksheet Title: BlankKrantz-05_3-
R8.mws     A Maple  Release 8 worksheet.\n\nAuthor:  Brian E. Blank " 
}}{PARA 0 "" 0 "" {TEXT -1 42 "Date Created:  8 September 2006 (Maple \+
R8)" }}{PARA 0 "" 0 "" {TEXT -1 482 "Date Last Revised: 8 September 20
06\n\nThis document may not be distributed by any medium,\nincluding p
rint, disk, and electronic transfer, without\nprior written permission
 of the author.\n\nFor more information,  please contact the author:\n
    \n     Department of Mathematics, \n     Washington University in \+
St. Louis\n     St. Louis, MO   63130\n   \n     Telephone:    (314) 9
35-6763\n             e-mail:    brian@math.wustl.edu\n\nCopyright:  \+
\251 2006 Brian E. Blank,  All Rights Reserved.\n" }}{PARA 3 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }