Engineering Applications in Differential and Integral ...

1 Engineering Applications in Differentialand Integral Calculus*ALAN HORWITZM athematics Department, Delaware County Campus, Penn State University, Pennsylvania, USAE-mail: EBRAHIMPOURC ollege of Engineering , Civil Engineering Program, Idaho State University, Idaho, Pocatello 83209, : authors describe a two-year collaborative project between the mathematics and the Engin-eering Departments. The collaboration effort involved enhancing the first year calculus courseswith applied Engineering and science projects. Two enhanced sections of the Differential (firstsemester) and Integral (second semester) calculus courses were offered during the duration of theproject. The application projects involved both teamwork and individual work, and we required useof both programmable calculators and Matlab for these projects.

2 Some projects involved use of realdata often collected by the involved faculty. The paper lists all the projects, including where they fitwithin the course topics. Some selected projects are described in detail for both the Differential andthe Integral calculus courses. The paper also summarizes the results of the survey questions given tothe students in two of the courses followed by the authors own critique of the enhancement PENN STATE, most of Math 140 coversdifferential calculus, while about 30% of thecourse is devoted to Integral calculus. Among thetopics covered are: limits and rates of change,continuous functions, derivatives of polynomials,rational functions, trigonometric functions, curvesketchingandoptimization,appliedwor dproblems, the Riemann Integral and the Funda-mental Theorem of Calculus, areas betweencurves, and volumes of solids of all of the topics covered in Math 141involve the Integral calculus including: inversefunctions, derivatives and integrals of exponentialand logarithmic functions, techniques of integra-tion, infinite sequences and series, parametricequations, and polar Fall 1997 to Spring 1999, we offeredenhanced sections of the Math 140 and Math141.

3 The objectives were:.to introduce team-based projects in engineeringand science,.to convey to the students the importance ofmathematics in Engineering and science,.to use Matlab and graphics calculators to ana-lyze experimental data and perform mathemati-cal offered one enhanced section of Math 140 inFall 1997 and also in Fall 1998. We also offeredone section of enhanced Math 141 in Spring 1998and in Spring 1999. In this paper, we will describethe projects used, the grading system, and surveyresults of the students' DESCRIPTIONS OF THE PROJECTSThe following are the projects (with pertinentmathematical topics in parentheses) we used in theenhanced section. In the next section, selectedprojects are described in 140 Projects, Fall 19971.

4 Data on strength of basswood samples (fittingdata with linear and polynomial functions)2. Temperature data from State College, Pennsyl-vania (fitting data with sine functions, compar-ing average and instantaneous rates of change)3. Beam analysis in mechanics (piecewise linear,quadratic, and cubic functions, and differen-tiability of a function)4. Electrical circuit analysis (exponential functionsand derivatives)5. Fitting a pipeline with minimal cost (optimiza-tion of a function on a closed interval)6. Crankshaft design (optimization of a functionon a closed interval)Math 141 Projects, Spring 19981. Analysis of beams in mechanics (polynomialintegration and optimization of a function ona closed interval)2.

5 Tuning a radio (integration of sine and cosinefunctions)3. Application of parametric curves (Cubic BezierCurves)4. Wrecking ball (approximating an integrand* Accepted 25 June J. Engng 18, No. 1, pp. 78 88, 20020949-149X/91 $ + in Great Britain.#2002 TEMPUS )Math140 Projects, (fittingdatawithpowerfunctionsandpropert iesoflogandexponentialfunctions) (piecewisefunctionsanddifferentiabilityo fafunction) (fittingpolynomialfunctionstovelocitydat aandtestingmodels) (planegeometry,trigonometry,andminimizat ionoffunctions).Math141 Projects, (exponentialfunctions,thedefiniteintegra landaveragevalueofafunction) (integra-tionofrationalfunctionsanduseof integrationtables) (fittingpolynomialfunctionstovelocitydat aandnumericalintegration) (CubicBezierCurves).

6 SELECTEDPROJECTSFROMFIRSTSEMESTERCALCULU SH ydraulicEngineering(Torricelli'sPrincipl e)Letfdenotethevolumeflowrateofaliquidth rougharestriction,suchasanopeningoravalv e, ' r ,trans-formthevariablesusinglogarithmsto base10first,givingalinearequationbetween thevariablesx log10 Vandb ,takenfrom[6],wascoveredinclass:` ,thefaucet'sflowratewasadjusteduntilthew aterlevelremainedconstantat15cups, 'Iftequalsthetimeforonecuptoflowoutofthe pot,thentheflowrateequals1= c1V c2tothegivendata, ,alongwiththedata, , , ,thetheoryisTorricelli'sprinciple,whichs aysthatf r ,forexperi-mentaldata,therelationshipisa pproximatelyf rV1= :0062V :0718, ,theMatlabcommandsare:L2 poly-fit(log10(cups),log10(flow),1),whic hgivesL2 , ;Som :4331,b 1 ,10logf 10b mx 10b10mx 10b 10logV m 10bVm rVm,wherer ,f `10.

7 ^( ).*v.^.4331',orf :0499 ( ) ( ).Foragivenopeningsize,thetimetofillacup ,t,wasrecordedforconstantwaterlevels(V 6;9;12;15cups) , [7]. (costofpipeof$ )andterraintype(normalterraininstallatio ncostof$ ).Installationinthewetlandrequiresanaddi tionalTrackHoeatacostof$60 ,theTrackHoecandigapproximately300feetof trench,andthusthereisanadditionalcostof$ 60=hr:30ft:=hr: $2= (cups)151296t(s)6789 EngineeringApplicationsinDifferentialand IntegralCalculus79 Thisgivesawetlandinstallationcostof$ (labeled1,2, ):Route1:Cost 2:7 d1 d2 d3 Route2:Cost 4:7 d1 d3 2 d22qRoute3:Cost 2:7x 4:7 d1 x 2 d22q,0 x d1 d3 Thesolutionforthethirdroutewasexplainedi ndetailinclassusinggeometry, Differential calculus, :Definevariable,x sym 0x0 Definecostfunction,C x Findthederivativeofthecostfunction,dC diff C Setthederivativeequaltozero,solveforx, , ,giventhecostsinvolved, , [6].

8 Figure4showsthecross-sectionofanirrigati onchannel, ,wewantthecross-sectionalarea,A,tobefixe d,tosayA ,L,ofthechannel'sperimeter, ,weassumedthat 1 2 ,h1 h2 h,ande1 e2 ,onecanexpressLasafunctionof 100d dtan 2dsin MinimizingLasafunctionofboth anddrequiresmultivariablecalculus, ,andminimizewithrespecttodExample: =3)L f d 100=d <d< d 0yieldsd 10= 34p 7 d 200=d3 >0on 0;1 ,f 7:5984 26 ,andminimizewithrespectto Example:d 1)L g 100 1=tan 2=sin 100 cot 2csc .Wewanttheglobalminimumfor0< < = cos 1 1=2 = =3 101 , ,however, ,thestudentscouldnotassumethat 1 2 ,h1 h2 h,ande1 e2 , 1,and 2andaskedto:.obtainaformulafortheperimet erL(excludingthefourthside);.minimizeLwi threspecttodon 0;1 ,andexplainwhyit'stheglobalminimum.

9 Usetheirgivenvalueof 1,alongwithd 2on 0; =2 ,andagainexplainwhyit' , , ,h, (temperaturedata) ,T,versustime,t, ;.predictthetemperatureusinginterpolatio nandextrapolation;. , ,fromt 0tot 0tot t cekt 24tothedata,andplottedthedataalongwithth egraphofT t . t :0358t 4:2668,whichgivesthetemperaturefunctionT t 71:2931e :0358t ,toinvolvethestudentwithunderstand-ingpr opertiesofexponentialfunctions, x definedonaclosedinterval[a;b],inintegral calculusonedefinestheaveragevalueofftobe 1b a baf x dx:Forthisproject,wewantedtheaveragevalu eofT t on[0;15],Tave 115 150 71:2931e :0358t 24 dt 79:1627 Finally,wecomparedTavewiththeaveragetem- peratureoverthetimeinterval[0,15]usingth edataonly, , ,andtheeffectofdragforceonfreefallveloci tyofaskydiver[4]providedagoodapplication ( ).

10 Amini-lecturewasgiventoexplaintherelatio n-shipbetweentheposition,velocity, ,FD cv2,wherec 0:2088Ns2=m2,determine(a)theterminalvelo city,vt,(b)thevelocityafter300moffreefal l;and(c)timefortheskydivertoreachaspeedo f160 (a),itwasnecessarytoexplainbrieflyNewton ' ,XF ma;mg FD ma;a 9:81 0:00348v2;a 0)vt 53:1m=s:Part(b)involvedthefollowingrelat ionamongacceleration,a,velocity,v,anddis placement,x:adx vdv;dx dv= 9:81 :00348v2 ; x 0dx vv 0 v=9:81 :00348v2 9:81 :00348v2givesv 53:094 1 e :0069xp;v300 53:094 1 e :00696 300 p 49:7 (c)oftheproblemusedthefollowingrelation: dt dv=a;since160km=h 44:44m=s,wehave t44:44t 0dt 44:44v 0dv= 9:81 :00348v2 .Makingasubstitutionandusingatableofinte gralsyieldst44:44 44:44v 0 dv=9:81 :00348v2 2:706ln v 53:0939=53:0939 v 44:440 2:706ln11:2705 6 :Taylorpolynomials,approx-imateintegrati on, [1].