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Click on a [-] sign to collapse a section. To do these exercises y ou will have to insert execution groups. That can be done by clicking \+ on the toolbar icon that looks like \"[>\". It can also be done via th e Insert menu." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 283 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", \+ you will be asked to use " }{TEXT 282 5 "MAPLE" }{TEXT -1 42 " to st udy some techniques of integration." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }{TEXT 276 5 "MAPLE" }{TEXT -1 156 " should be prepared with t he same care that you would devote to laboratory reports in biology an d chemistry. A report should not be a diary or history of a" }{TEXT 259 1 " " }{TEXT 277 5 "MAPLE" }{TEXT -1 77 " session. Delete what is not needed for the report. All lines of the form " }{TEXT 260 6 "?t opic" }{TEXT -1 74 " should be erased. All errors should be erased. \+ When you are printing a" }{TEXT 261 2 " " }{TEXT 278 5 "MAPLE" } {TEXT -1 113 "report, think about the toner and paper resources that y ou are using. When you assign a variable, for example \"" }{TEXT 262 8 " x := 5;" }{TEXT -1 29 " \", there is no need to have" }{TEXT 263 1 " " }{TEXT 279 5 "MAPLE" }{TEXT 280 1 " " }{TEXT -1 11 "echo bac k " }{TEXT 264 6 "x := 5" }{TEXT -1 89 ". When this is printed, it s imply wastes paper and ink. Choose the silent terminator \"" }{TEXT 265 9 " x := 5: " }{TEXT -1 38 " \" instead. When you load a package ," }{TEXT 266 2 " " }{TEXT 281 5 "MAPLE" }{TEXT -1 95 " may list the commands that become available with the package. Delete these before \+ printing. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Some of the text in this worksheet should be deleted. Delete \+ the sections the " }{TEXT 267 12 "Introduction" }{TEXT -1 5 " and " } {TEXT 268 8 "Keywords" }{TEXT -1 34 " sections. Delete this section on " }{TEXT 269 7 "Reports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your worksheet shou ld execute in the order that it has been written. In particular, reme mber that the ditto refers to the result of the last executed command \+ - not the result of the command that physically precedes it. " }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Key Words" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT -1 34 "On-line informat ion about using a " }{TEXT 275 5 "MAPLE" }{TEXT -1 199 " function can \+ be obtained at a prompt by preceding the function by a question mark a nd executing that line. For example, execute the next line to see the \+ information on the differentiation function " }{MPLTEXT 1 0 4 "diff" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?diff" }} }{EXCHG {PARA 272 "" 0 "" {TEXT -1 27 "For help on \"help\", enter " }{MPLTEXT 1 0 2 "??" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "??" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 152 "The wor ds and symbols that are used in this worksheet have been collected her e as hypertext links. Click on any word to go to the correspond help p age." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "changev ar" 2 "changevar" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "help" 2 "?" 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The amount of gold contained in the bead can be \+ computed from the volume of the bead. The bead has the shape of a soli d sphere of radius " }{TEXT 286 1 "R" }{TEXT -1 37 " from which a c ylinder of radius " }{TEXT 287 1 "r" }{TEXT -1 181 " has been bored . \n\n(The spherical caps of the cylinder are of course excluded too. \+ The easiest way to picture the solid is to first imagine a solid spher e. A drill bit of radius " }{TEXT 288 1 "r" }{TEXT -1 121 " is pla ced on the surface of the sphere and the center of the drill follows a diameter through the sphere.) \n\nUse the " }{TEXT 344 13 "Washer Me thod" }{TEXT -1 86 " to calculate the volume of the resulting surface . Your answer will depend on both " }{TEXT 342 1 "r" }{TEXT -1 8 " \+ and " }{TEXT 343 1 "R" }{TEXT -1 4 ". 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::J::1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 13 "Exercise 2 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "A fla shlight reflector is made of an aluminum alloy whose mass density is \+ " }{TEXT 297 1 " " }{XPPEDIT 298 0 "3.743*g/cm^3" "6#*($\"%VP!\"$\"\" \"%\"gGF'*$%#cmG\"\"$!\"\"" }{TEXT 299 1 " " }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "The shap e of the reflector is essentially a paraboloid of revolution. To be ex act, the reflector occupies the solid region that is obtained when the region bounded by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 18 " " }{TEXT 291 13 " " } {TEXT 300 1 " " }{XPPEDIT 292 0 "y = 2.530*x^(1/2)" "6#/%\"yG*&$\"%ID! \"$\"\"\")%\"xG*&\"\"\"F)\"\"#!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 38 " " }{TEXT 293 1 " " }{TEXT 301 1 " " }{XPPEDIT 294 0 "y = 2.530*x^(1/2)+` 0.300`" "6#/%\"yG,&*&$\"%ID!\"$\"\"\")%\"xG*&\"\"\"F*\"\"#!\"\"F*F*%&0 .300GF*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ " }{TEXT 295 1 " " }{XPPEDIT 296 1 "`0 < x < 2.5 cm`" "6#%/0~<~x~<~2.5~cmG" }{TEXT -1 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 0 {PARA 4 "" 0 " " {TEXT -1 42 " a) Plot a cross-section of the reflector." }}{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 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 39 " b) What is the mass of the reflector?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 12 "Exercise \+ 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "A \+ cylindrical tank has height " }{XPPEDIT 302 0 "L" "6#%\"LG" }{TEXT -1 15 " and radius " }{TEXT 303 1 " " }{XPPEDIT 304 0 "r" "6#%\"rG " }{TEXT -1 53 ". The tank has a circular base but no top. It is \+ " }{TEXT 305 10 "partially " }{TEXT -1 24 " filled with a volume " } {XPPEDIT 306 0 "V" "6#%\"VG" }{TEXT 307 1 " " }{TEXT -1 70 " of a fl uid. The axis of symmetry of the tank is along the positive " }{TEXT 311 1 "y" }{TEXT -1 50 "-axis and one diameter of its base is along th e " }{TEXT 309 1 "x" }{TEXT -1 46 "-axis. When the tank is revolve d about the " }{TEXT 310 1 "y" }{TEXT -1 29 "-axis with angular spee d " }{XPPEDIT 308 1 "omega" "6#%&omegaG" }{TEXT -1 100 " , the su rface of the fluid assumes a shape that is the paraboloid of revolutio n that results when" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 " " }{TEXT 312 1 " " }{XPPEDIT 313 1 "y = \+ h+omega^2*x^2/(2*g);" "6#/%\"yG,&%\"hG\"\"\"*&*&)%&omegaG\"\"#F')%\"xG \"\"#F'F'*&\"\"#F'%\"gGF'!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{XPPEDIT 314 1 "`0 < x < r`" "6#%*0~<~x~<~rG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Here \+ " }{XPPEDIT 317 1 "h" "6#%\"hG" }{TEXT 318 1 " " }{TEXT -1 21 " is so me number in " }{XPPEDIT 315 1 "[0, V/Pi/r^2]" "6#7$\"\"!*(%\"VG\"\" \"%#PiG!\"\"*$%\"rG\"\"#F)" }{TEXT 316 1 " " }{TEXT -1 19 " that depe nds on " }{XPPEDIT 319 1 "omega" "6#%&omegaG" }{TEXT -1 5 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "(The for mula above is valid for angular speeds at which the surface of the flu id has not yet touched the base or the mouth of the tank.)" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 " a) Find " }{XPPEDIT 320 1 "h" "6#%\" hG" }{TEXT 321 1 " " }{TEXT -1 22 " as a function of " }{XPPEDIT 322 1 "omega" "6#%&omegaG" }{TEXT -1 24 " (for angular speeds " } {XPPEDIT 323 1 "omega" "6#%&omegaG" }{TEXT -1 34 " at which there is no spilling)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 4 "" 0 "" {TEXT -1 20 " b) At what value " }{XPPEDIT 326 1 "omega[S]" "6#&%&omegaG6#%\" SG" }{TEXT 327 1 " " }{TEXT -1 8 " of " }{XPPEDIT 324 1 "omega" "6 #%&omegaG" }{TEXT 325 1 " " }{TEXT -1 38 " does spilling begin, assum ing that " }{XPPEDIT 328 1 "h(omega)" "6#-%\"hG6#%&omegaG" }{TEXT -1 36 " has not reached 0 for values of " }{XPPEDIT 329 0 "omega" "6#% &omegaG" }{TEXT -1 13 " less than " }{XPPEDIT 330 1 "omega[S]" "6#&% &omegaG6#%\"SG" }{TEXT -1 3 " ?" }}{PARA 4 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 4 "" 0 "" {TEXT -1 28 " c) At what angular s peed " }{XPPEDIT 331 1 "omega[B]" "6#&%&omegaG6#%\"BG" }{TEXT 332 1 " " }{TEXT -1 125 " does the surface of the fluid touch the bottom of \+ the tank, assuming that spilling has not occurred for smaller values o f " }{TEXT 333 1 " " }{XPPEDIT 334 1 "omega" "6#%&omegaG" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "d) Discuss the question: As \+ " }{TEXT 335 1 " " }{XPPEDIT 336 1 "omega" "6#%&omegaG" }{TEXT -1 101 " increases, does the surface of the fluid touch the base of the tank or the mouth of the tank first?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Exercise 4 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The equat ion of the Gateway arch is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 337 1 " " }{XPPEDIT 338 0 "y = 6 93.8597-68.7673*cosh(.100333e-1*x)" "6#/%\"yG,&$\"((fQp!\"%\"\"\"*&$\" 'two!\"%F)-%%coshG6#*&$\"'L.5!\"(F)%\"xGF)F)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 339 2 " " }{XPPEDIT 340 0 "`-299.2239 < x < 299.2239`" "6#%9-299.2239~<~x~< ~299.2239G" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 114 "Find an equation of a parabola with the same vertex and \+ feet. Plot both curves on the same set of coordinate axes." }}{PARA 0 "" 0 "" {TEXT -1 27 "Compute the length of each." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Informa tion" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 46 "M132-6R5.mws A Maple V Release 5 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank 26 February 1998 " }} {PARA 0 "" 0 "" {TEXT -1 31 "Last Revision: 25 February 1999" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, without" }}{PARA 0 " " 0 "" {TEXT -1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more info rmation, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Lo uis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " T elephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " \+ e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 " " }}{PARA 269 "" 0 "" {TEXT -1 61 "Copyright: \251 1997-1999 Brian E. Blank, All Rights Reserved." }}}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 3 4 1802 }