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0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 59 "In this section we illustrate how to use Maple to compute \n" }{TEXT 212 35 "1. areas bounded by curves.\n" } {TEXT 212 40 "2. volume of solid of revolution." }{TEXT 212 1 " \n" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 210 4 "Ar ea" }{TEXT 212 14 ", for example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=0.3*x^3-3.3*x^2+9.6*x+3.33;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=-0.15*x^2+2.03*x+3.33;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 78 "To compute the a reas bounded by the graph of f and the graph of g over [1,8],\n" } {TEXT 212 60 "We plot f and g on the same coordinate axes with the com mand" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(\{f,g\},x=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 74 "The plot shows two poin ts of intersection, which can be found with command" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rt:=fsolv e(f=g,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 204 "between x =1 and x = rt[2] the graph of g is below the \+ graph of f, over [rt[2], rt[3]] the graph of f is below the graph of g and on [rt[3], 8] the graph of g is below the graph of f , therefore \+ the area\n" }{TEXT 212 18 "can be computed by" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "A:=Int(f-g,x=1. .rt[2])+Int(g-f,x=rt[2]..rt[3])+Int(f-g,x=rt[3]..8); value(A);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 211 6 "Volume" }{TEXT 212 13 ", for example" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=-x^2+5*x-2;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "g:=x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "h:=-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 92 "Find the volume of the solid obtained by rotating the region bounded by f and g about y=-1.\n" }{TEXT 212 38 "Plot f , g and y=-1 on the same plane\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(\{f,g,h\},x=0..4);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 39 "Two points of intersection can fo und by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sl:=fsolve(f=g,x) ; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 177 "The cross section of this solid is a washer, the radius of th e big disk is f+1 and the radius of the small disk is x+1. The volume \+ of this solid is easily accomplished in Maple." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "B:=Int(Pi* ((f+1)^2-(g+1)^2),x=sl[1]..sl[2]); value(B);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 212 123 "Now find the volume of the solid obtained by rotating this region about the Y-axis, using the method of cylindrical shells. " }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "C:=Int(2*Pi*x*(f-g),x=sl[1]..sl[2]); value(C);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 77 "Now we w ill do an excercise whose volume can be computed by both the methods." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:=3*x;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:=x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(f=g,x);plot(\{f,g\},x=0..4);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 118 "Find the volume of the solid obt ained by rotating the region bounded by f and g about the Y-axis, by b oth the methods." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:=Int( Pi*(sqrt(y)^2-(y/3)^2),y=0..9); value(A);" }{MPLTEXT 1 0 38 "B:=Int(2* Pi*x*(f-g),x=0..3); value(B);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 107 "1. Now, you fin d the volume, using BOTH the methods, when the above region between y \+ = 3*x and y = x^2 is " }{TEXT 212 29 " rotated about the X-axis... " }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }} {PARA 0 "" 0 "" {TEXT 212 72 "2. Find the area of the region that is b ounded above by y = 7ln(x) and " }{TEXT 212 13 "4- (x^3) -x " }{TEXT 212 25 "and below by the x-axis.\n" }{TEXT 212 1 "3" }{TEXT 212 40 ". Consider the region in the exercise 1\n" }{TEXT 212 89 "a) Fin d the volume of the solid obtained by rotating the region about the x- axis.\n" }{TEXT 212 89 "b) Find the volume of the solid obtaine d by rotating the region about the y-axis.\n" }{TEXT 212 102 "c) \+ Find the volume of the solid obtained by rotating the region about t he vertical line x = 4.\n" }{TEXT 212 104 "d) Find the volume o f the solid obtained by rotating the region about the horizontal line \+ y = 4.\n" }{TEXT 212 1 "4" }{TEXT 212 67 ". Compute the volume of the \+ solid obtained by rotating the ellipse " }{TEXT 212 23 "x^2/a^2 + y^2/ b^2 = 1 " }{TEXT 212 53 " \+ \n" }{TEXT 212 24 " about the x-axis.\n" }{TEXT 212 1 "5" }{TEXT 212 109 ". A circular doughnut is formed by rotating a circle \+ of radius r centered at x = a, y = 0 about the y- axis.\n" }{TEXT 212 59 "a) Find the equation of the circle described above.\n" } {TEXT 212 88 "b) Use cylindrical shell technique to find a form ula for the volume of doughnut.\n" }{TEXT 212 72 "c) Evaluate t he volume in the special case where a = 3 and r = 1." }}{PARA 0 "" 0 " " {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }