Fwd: RE: Antiderivatives and Definite Integrals
- To: mathgroup at smc.vnet.net
- Subject: [mg39841] Fwd: [mg39713] RE: [mg39670] Antiderivatives and Definite Integrals
- From: Garry Helzer <gah at math.umd.edu>
- Date: Sat, 8 Mar 2003 02:50:48 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Begin forwarded message: > From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com> To: mathgroup at smc.vnet.net > Date: Mon Mar 3, 2003 11:48:37 PM US/Eastern > To: mathgroup at smc.vnet.net > Subject: [mg39841] [mg39713] RE: [mg39670] Antiderivatives and Definite Integrals > > >> -----Original Message----- >> From: Garry Helzer [mailto:gah at math.umd.edu] To: mathgroup at smc.vnet.net > To: mathgroup at smc.vnet.net >> Sent: Saturday, March 01, 2003 8:48 AM >> To: mathgroup at smc.vnet.net >> Subject: [mg39841] [mg39713] [mg39670] Antiderivatives and Definite Integrals >> >> >> The antiderivative of Sqrt[1+Cos[x]] discussed here recently (sorry, I >> lost the thread) provides an amusing illustration of the fact that >> Mathematica does not always evaluate definite integrals by first >> finding an antiderivative and then substituting in the upper and lower >> limits. (See the Mathematica book A.9.5) Make the definitions >> >> f[x_] = Integrate[Sqrt[1 + Cos[t]], {t, 0, x}] >> g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}] >> >> Then f[2Pi] is 0 (wrong) and g[2Pi] if 4Sqrt[2] (correct). Of course >> f[x]==g[x] returns True. >> >> Garry Helzer >> Department of Mathematics >> University of Maryland >> 1303 Math Bldg >> College Park, MD 20742-4015 >> >> > > You might be interested to observe: > > > In[1]:= f[x_] = Integrate[Sqrt[1 + Cos[t]], {t, 0, x}] > Out[1]= 2*Sqrt[1 + Cos[x]]*Tan[x/2] > > In[2]:= g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}] > > In[3]:= ?g > Global`g > g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}] > > In[4]:= Plot[f[x], {x, 0, 2Pi}] // Timing > Out[4]= {0.03 Second, = Graphics =} > > ...what you called "wrong". Well, it is wrong. From calculus we know that 1. A function defined as an integral with a variable upper limit is an antiderivative. 2. An antiderivative is, by definition, differentiable. 3. A differentiable function is continuous. So, by this plot, f[x] is not the antiderivative of any function. But this may be too rigid a view. After all, D[f[x],x]//FullSimplify brings you back to the integrand. This is a formal computation that misses the points of discontinuity. Perhaps the algorithms used by Mathematica are guaranteed only to produce such formal antiderivatives. A superficial look a the Risch algorithm suggests that this might be the case since it proceeds by formal manipulation in extension fields of fields of rational functions. As I was typing this a stranger wandered in looking for an arc length(it happens). His problem provides another example. Let r=2-Sin[t] and try Integrate[Sqrt[r^2 +D[r,t]^2],{t,0,Pi/2}]//N. The result is about -15, but arc length should not be negative. Set h[x_]= Integrate[Sqrt[r^2 +D[r,t]^2],{t,0,x}] and again we get a formal antiderivative , the plot of which reveals discontinuities. It cannot be carelessly used to evaluate definite integrals. (If you simplify Sqrt[r^2 +D[r,t]^2] before integrating you get an honest antiderivative.) In[5]:= > Plot[g[x], {x, 0, 2Pi}, PlotRange -> {All, {0, 6}}] // Timing > Out[5]= {4.096 Second, = Graphics =} > > ...what you called "right". It is right. But the corresponding function for the arc length problem is wrong. > > > Redefine: > > In[6]:= f[x_] /; -Pi < x < Pi = f[x] > Out[6]= 2*Sqrt[1 + Cos[x]]*Tan[x/2] > > In[7]:= f[x_] /; Pi < x < 3 Pi = f[x] + 4*Sqrt[2] > Out[7]= 4*Sqrt[2] + 2*Sqrt[1 + Cos[x]]*Tan[x/2] Or 4Sqrt[2]Round[x/(2Pi)]+ 2*Sqrt[1 + Cos[x]]*Tan[x/2] . But these formulas are less than perfect since they are indeterminate at odd multiples of Pi. > > > In[9]:= > Plot[f[x], {x, 0, 2Pi}, PlotRange -> {All, {0, 6}}] // Timing > Out[9]= {0.03 Second, = Graphics =} > > Don't be confused by that double use of f (as a function, and as a > shortcut > for an algebraic expression). > > -- > Hartmut Wolf > > > Garry Helzer Department of Mathematics University of Maryland 1303 Math Bldg College Park, MD 20742-4015
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