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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "working with integrals" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "Being able to work with integra
ls involving parameters and change variables is the most important asp
ect of integrals to have mastered. Technology can do the actual findin
g of antiderivatives for evaluating purposes. " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Consider the following no
nnegative function on the interval from 0 to " }{XPPEDIT 18 0 "a;" "6#
%\"aG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1
26 " is a positive parameter: " }{XPPEDIT 18 0 "0 < a;" "6#2\"\"!%\"aG
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(x^3
*sqrt(a^2-x^2),x);\nvalue(%);\nInt(x^3*sqrt(a^2-x^2),x=0..a);\nvalue(%
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The complex sign of a posit
ive number is just one: " }{XPPEDIT 18 0 "csgn(a) = 1;" "6#/-%%csgnG6#
%\"aG\"\"\"" }{TEXT -1 51 ". We can get rid of this annoying sign by a
ssuming " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 90 " is positive (wh
ich puts a tilde after the variable name to remind us of this assumpti
on):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "assume(a>0);\nInt(x
^3*sqrt(a^2-x^2),x=0..a)\n=int(x^3*sqrt(a^2-x^2),x=0..a);" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The param
eter " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 33 " sets the scale for
the variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 139 " and must
have the same units in a physical application for the units to combin
e correctly in the integral expression. Therefore the ratio " }
{XPPEDIT 18 0 "z = x/a;" "6#/%\"zG*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1
81 " must be a dimensionless variable, which is equivalent to measurin
g the variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 13 " in units
of " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 8 ", i.e., " }{XPPEDIT
18 0 "z;" "6#%\"zG" }{TEXT -1 22 " has the value 2 when " }{XPPEDIT
18 0 "x = 2*a;" "6#/%\"xG*&\"\"#\"\"\"%\"aGF'" }{TEXT -1 13 ". By sett
ing " }{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\"" }{TEXT -1 40 " we can s
pecialize to the case in which " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT
-1 120 " is directly measured in these units without actually changing
the variable, and we are then able to plot the function. " }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "In particular, b
y dividing the function by its length, we get a new nonnegative functi
on with unit area on this same interval, so it can serve as a probabil
ity distribution function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
79 "Int(15/2/a^5*x^3*sqrt(a^2-x^2),x=0..a)\n=int(15/2/a^5*x^3*sqrt(a^2
-x^2),x=0..a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Here is what it
looks like in dimensionless units:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 5 "a:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pl
ot(15/2/a^5*x^3*sqrt(a^2-x^2),x=0..a);" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 137 "Suppose this function represents the probability density
that a self-serve filling station customer will fill his or her car's
tank with " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 36 " gals of gas,
with maximum capacity " }{XPPEDIT 18 0 "a = 12*gals;" "6#/%\"aG*&\"#7
\"\"\"%%galsGF'" }{TEXT -1 105 ". It is clearly much more probable tha
t a large fraction of the tank will be added than a small fraction." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "The probability that the custom
er will fill the tank no more than half full is the integral from 0 to
half the tank capacity " }{XPPEDIT 18 0 "a = 12*gals;" "6#/%\"aG*&\"#
7\"\"\"%%galsGF'" }{TEXT -1 21 " is about 10 percent:" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "a:='a': assume(a>0);\nint(15/2/a^5*
x^3*sqrt(a^2-x^2),x=0..a/2);\nevalf(%);" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 107 "The probability that the tank will be filled to more tha
n three fourths capacity is instead just over half:" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 53 "int(15/2/a^5*x^3*sqrt(a^2-x^2),x=3/4*a..a
);\nevalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The expected val
ue of the number of gallons added to the tank is:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 "int(15/2/a^5*x^4*sqrt(a^2-x^2),x=0..a);\nmu:
=evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "which for a 12 gal \+
tank is just:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mu := eval
f(.7363107783*12*gals,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "On t
he other hand the dimensionless variable " }{XPPEDIT 18 0 "z = x/a;" "
6#/%\"zG*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 123 " represents the fracti
on of the tank's full capacity that is added to the tank, which is abo
ut three fourths its capacity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "We have not directly learned a te
chnique for evaluating this expected value integral but Maple easily f
inds an antiderivative involving an inverse trig function which can be
evaluated exactly, though only numerical values make sense in this ap
plication:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(15/2/a^5*
x^4*sqrt(a^2-x^2),x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "By set
ting " }{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\"" }{TEXT -1 91 " again w
e can directly find the median of the distribution in the fractional t
ank variable " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 0 "" }{TEXT -1
85 " (half the people fill up the tank to this fraction) by putting in
an extra power of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 17 " in t
he integral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "a:=1;\nint(
15/2/a^5*x^4*sqrt(a^2-x^2),x=0..X)=1/2;\nz_median:=fsolve(%,X=0..1);\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "On the average with this mod
el, most people put additional gas amounting to about 87 percent of fu
ll capacity, while the expected fill fraction is about 3/4 of a tank f
or the average person who stops to add gas." }}}}{MARK "0 0 0" 0 }
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