{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE " " -1 -1 "Helvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Couri er" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "The Area between Two Curv es" }}{PARA 0 "" 0 "" {TEXT -1 55 "Example 1. Find the area enclosed \+ by y=f(x) and y=g(x)" }}{PARA 0 "" 0 "" {TEXT -1 197 "Solution: The \+ first step in a problem which involves finding the area bounded betwee n two curves is to plot the graph of the two curves simultaneously. U sing Maple V we define the two functions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := x -> 4 - x^2; g := x->exp(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now plot both curves simultaneously." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{f(x),g(x)\},x=-5..5,y=-10..10);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol := fsolve(\{y=f(x),y=g (x)\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:=fsolv e(f(x)-g(x)=0,x,-3..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " b:=fsolve(f(x)-g(x)=0,x,0..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "Observe that the points of intersection have coordinates given by (-1,3), and (3,-5), which agrees with the previous hand calculatio n. The value of the integral is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Int(f(x)-g(x),x=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Exam ple 2. Find the area of bounded by x=cos(y) and x=cos(2*y)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "implicitplot(\{x=cos(y),x=2*cos(y) \},x=-1..2,y=-Pi..Pi);" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(2*cos(y)-cos(y),y=-Pi/2..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 " Remark: Of course if we write x=y in the question above, we get an are a between y=cos(x) and y=2*cos(x), which will use 'vertical' rectangle to find the area." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "15" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }