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A simple algorithm to implement the trapezoidalrule for unequally spaceddata can be variand written as in Fig 1714Two vectors,x andy, holding the independent dependent that (a) the two vecinto the M-file Two error trapsareincludedto ensure ablesarepassed orderrA loop is employedto tors are of the samelength and (b) the x's are in ascending from thoseof Eq (1730) generate integralNotice that we havemodified the subscripts the in subscripts anays to accountfor the fact that MATLAB doesnot allow zero for An applicationof the M-file can be developed the sameproblemthat was solvedin E x a m p l e1 7 6 :
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>> x - l0 I2 22 32 >> v - 42+25*x-200*x > > l -r a p L l n e q ( x , y ) Bl; 36 44454 641 ' 2 + 5 15 * x ^ 3 - 9 0 0 " x ^ 4 + 4 0 0 * x " 5 ;
which is identical the resultobtained Example176 in to
F I G U R E7 1 4 1 the rule spoced dofo M{ileio lmplement tropezoidol lorunequolly
function
o o ir_n ru J
= trapuneq(x,y)
J,--4)a L -r-i, >lJdLFu :d L r a p s T u ', v ( , r_ l ,tr-L o t dlt:alt_ftte YJ
T = trapuneq (x, y) : Z rule to determine the integral Applies the trapezoidal (x, y) where x and y must be of t,he * for n daLa points ascending % same lengrth and x must be monotonically % input: x = /ector of independent var:rables % y = vector of dependenE variai:les % output: 2 I=inteqralestimate least 2 input arguments resuired'),end if nargin<2,error('at not monotonically ascendrnq' ),end if any(diff (x)<0),error('x n = lengrth(x); if lengch (y) -=n, error ( 'x and y must be same length' ) ; end s = 0; for k = 1;n-1 +y\k+\\) /2; s : s + (x(k+1)-x(k))-(y(k) end
T c '
' The
r : l u n c t i o n s J e s e r i h eiJ S c c t i o r1 9 7I i n r
FORMULAS NUMERICAL INTEGRATION
MATTAB Functions: trapz
cumcrapz
MATLAB has a built-in function that evaluates integralsfor datain the same fashion asthc M-file we just presented Fig i714 It hasthe general in syntax
z t rapz (x, l')
wherethe two vectors,x &od-r, hold the independent dependent and variables, respectively Here is a simple MATLAB sessionthat uses this function to integratethe data from T a b l e1 7 3 : >> x = t0 I2 22 32 3,h 4 44 54 64 7 81; > > y = 0 2 + 2 5 * x - 2 0 0 " x " ' 2 + 6 1 * x ^ 3 - 9 0 0 * x ^ 4 + 4 0 0 * x ^ 5 ; 5
>> trapz (x,y)
15948 In addition, MAILAB has another function, cumr rapz that computes the cumulativc inte_ural sirnple representationof its syntax is A z cumtrapz (x, y)
where the two vectors, x and y, hold the independent and dependent variables,respectively, a n d z : a v e c t o r w h o s e e l e n r e n t s ( k ) h o l d t h e i n t e g r a lf r o r n x ( 1 ) t o x ( k ) z
EXAMPLE l77
UsingNumericol Integrotion Compute to Distonce fromVelociiy ProblemStotement As described the beginning this chapter, niceapplication at of a of integration to compute distance of an objectbased its velocity as is the u(t) in on z(r) (recall 172): Eq :trl: I u \ t ld t
Suppose that we had measurements velocityat a series discrete of of unequally spaced tirno during free fall Use Eq (172) to syntheticallygeneratesuch inforrnation a 70-kg for jumper with a drag coefficient of 0275 kg/m Incorporate some randomerrorby rounding the velocitiesto the nearest integer Then usecum1, rapz to determine distance and the fallen comparethe resultsto the analyticalsolution(Eq 17a)In addition,develop plotofth a analyticalarrd cornputed distances along with velocity on the samegraph Solution Someunequally spaced timesandrounded velocities be generated as can
' lornot shu'r (J ;';5=[Q 1 L4 2 3 43 5 51 B); >> 9-9 8L;m=70; cct=O 2'15 ; > > v = r o u n d ( s q r t ( g * m , / c d )* t a n h ( s q r t ( g * c d / m ) * t ) ) ; The distances can then be comouted as
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