Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Wednesday, March 5, 8.4 Infinite functions p445: 8,14,16,42 Main Idea: Watch out for embedded thunderstorms. Watch out for embedded tornados. Key Words: Unbounded functions. Goal: Learn how to compute area where the function is unbounded. -------------------------------------------------------- Previous assignment: Inf. Limits p441: (8),(14),(26),(36) Page 441 Problem 8 Infinity x INT -------- dx x=10 2 1+x x=b x INT -------- dx x=10 2 1+x _ 2 _ _ _ | ln(1+x ) | x=b | 2 | | ---------- | = 1/2 |ln(1+b )-ln(101) | = +Infinity |_ 2 _| x=10 |_ _| ----------------------------- Get["font.math"]; f[x_] := x/(1+x^2); P1 = Plot[f[x],{x,10,100}]; P2 = ParametricPlot[{10,y},{y,0,f[10]},PlotStyle->{RGBColor[1,0,0]}]; P3 = Show[P1,P2,PlotLabel->"P441 P8 y = x/(1+x^2)"]; Display["p8.ps",P3]; -------------------------------------------------------- Page 441 Problem 14 Infinity -x INT x e dx x=1 -x x -e -x dx e dx -x -x -x e + INT e dx _ _ | | x=b | -x -x | | -x e - e | |_ _| x=1 -b -b -b e - e + 1/e + 1/e = 2/e <== answer ------------------------------------------------------------ Get["font.math"]; f[x_] := x E^(-x); P1 = Plot[f[x],{x, 1,10}]; P2 = ParametricPlot[{1,y},{y,0,f[1]},PlotStyle->{RGBColor[1,0,0]}]; P3 = ParametricPlot[{0,y},{y,0,f[1]}]; P4 = Show[P1,P2,P3,PlotLabel->"P441 P14 y = x e^(-x)",PlotRange->All]; Display["p14.ps",P4]; ------------------------------------------------------------ Page 441 Problem 26 Find the area under the curve 1 y = -------- 2 x + x To the right of x = 1. 1 -1 y = ----------- + ------------ x x+1 _ _ | | x=b | ln(x) - ln(x+1) | |_ _| x=1 _ _ _ _ | | x | | x=b | ln| --------- | | |_ |_ x + 1 _| _| x=1 b 1 ln(--- ) - ln(---) = ln(2) b+1 2 ------------------------------------------------------------ Get["font.math"]; f[x_] := 1/(x^2+x); P1 = Plot[f[x],{x,1,10},PlotPoints->100]; P2 = ParametricPlot[{1,y},{y,0,f[1]},PlotStyle->{RGBColor[1,0,0]}]; P3 = ParametricPlot[{0,y},{y,0,f[1]}]; P4 = Show[P2,P1,P3,PlotLabel->"P441 P26 y = x/(x^2+x)",PlotRange->All]; Display["p26.ps",P4]; ------------------------------------------------------------ Page 441 Problem 36 In electromagnetic theory, the magnetic potential u at a point on the axis of a circular coil is given by Infinity dx u = A r INT -------------- x = a 2 2 3/2 (r + x ) where A, r, and a are constants. Evaluate u. /| / | / | / | 2 2 / | Sqrt[r + x ] / | r / | / | /t | /_________| x x = r Ctn[t] 2 dx = -r Csc [t] dt 2 2 Sqrt[r + x ] = r Csc[t] 2 Inf -r Csc [t] dt u = A r INT -------------- x=a 3 3 r Csc [t] Inf u = A/r INT -Sin[t] dt x=a _ _ x=Inf | | u = A/r | Cos[t] | |_ _| x=a _ _ | a | u = A/r | 1 - ----------- | | 2 2 | |_ Sqrt[a + r ] _| ------------------------------------------------------------ A r Integrate[(x^2+r^2)^(-3/2),{x,a,Infinity}] ------------------------------------------------------------ New Material: x=2 dx Page 443 Example 1: INT ------------ x=0 2 Sqrt[4-x ] x=2 dx INT -------------------- x=0 2 2 Sqrt[1-(x/2) ] _ _ x=2 1/2 dx | | x=2 INT ---------------- = | ArcSin[x/2] | = Pi/2 x=0 Sqrt[1-(x/2)^2] |_ _| x=0 --------------------------------------------------------- Get["font.math"]; f[x_] := 1/Sqrt[4-x^2]; P1 = Plot[f[x],{x,0,1.95}]; P2 = ParametricPlot[ {2,y},{y,0,f[1.95]},PlotStyle->{RGBColor[1,0,0]}]; P3 = Show[P1,P2,PlotLabel->"P443 Ex 1 y = 1/Sqrt[4-x^2]", AspectRatio->Automatic]; Display["ex1.ps",P3]; --------------------------------------------------------- x=16 1 Page 443 Example 2: INT -------- dx x=0 1/4 x _ _ | 3/4 | x=16 | x | 3/4 | --------- | = 32/3 - 4/3 a = 32/3 <== Answer. |_ 3/4 _| x=a --------------------------------------------------------- Get["font.math"]; f[x_] := x^(-1/4); P1 = Plot[f[x],{x, 16^(-4), 16},PlotPoints->10000, PlotStyle->{Thickness[0.005]}]; P2 = ParametricPlot[ {16,y},{y,0,f[16]}, PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}]; P3 = ParametricPlot[ {0,y},{y,0,16}, PlotStyle->{RGBColor[1,0,0],Thickness[0.001]}]; P4 = Show[P1,P2,P3,PlotLabel->"P443 Ex 2 y = 1/x^(1/4)", PlotRange->All,AspectRatio->Automatic]; Display["ex2.ps",P4]; ------------------------------------------------------------------- x=1 1 Page 443 Example 3: INT ---- dx x=0 x | x=1 ln(x) | = ln(1) - ln(a) = + Infinity | x=a --------------------------------------------------------- Get["font.math"]; f[x_] := 1/x; P1 = Plot[f[x],{x,0.01,1},PlotPoints->10000]; P2 = ParametricPlot[ {1,y},{y,0,f[1]},PlotStyle->{RGBColor[1,0,0]}]; P3 = ParametricPlot[ {0,y},{y,0,f[0.01]},PlotStyle->{RGBColor[1,0,0]}]; P4 = Show[P1,P2,P3,PlotLabel->"P421 Ex 3 y = 1/x",PlotRange->All]; Display["ex3.ps",P4]; ------------------------------------------------------------------- Page 443 Example 4: x=1 1 Show that INT ----- dx converges if p < 1 but diverges if p >= 1. x=0 p x ------------------------------------- Get["font.math"]; P1 = Table[ Plot[1/x^(p/10),{x,(1/2)^(10/p),2}, PlotStyle->{RGBColor[1,0,0]}],{p,1, 9}]; P2 = Table[ Plot[1/x^(p/10),{x,(1/2)^(10/p),2}, PlotStyle->{RGBColor[0,1,0]}],{p,10,20}]; P3 = Plot[0,{x,0,2}]; P4 = Show[P1,P2,P3, PlotLabel->"Page 443 Ex 4 INT 1/x^p, Red Converge; Green Diverge", PlotRange->All]; Display["ex4.ps",P4]; ------------------------------------ 2/3 2/3 Page 443 Example 5 x + y = 1 --------------------------------------------------------- Get["font.math"]; f[x_] := (1-Abs[x]^(2/3))^(3/2); g[x_] := -(1-Abs[x]^(2/3))^(3/2); P1 = Plot[f[x],{x,-1,1},PlotStyle->{Thickness[0.01]}]; P2 = Plot[g[x],{x,-1,1},PlotStyle->{Thickness[0.01]}]; P3 = Show[P1,P2,PlotLabel->"P443 Ex 5 x^(2/3) + y^(2/3) = 1", AspectRatio->Automatic]; Display["ex5.ps",P3]; l[a_,b_] := 3/2 (b^(2/3)-a^(2/3)); n[a_] := ((2/3)(1/4) + a^(2/3))^(3/2); A = Table[0,{i,1,10}]; Do[A[[i+1]] = n[A[[i]]],{i,1,9}]; B = Table[{A[[i]],f[A[[i]]]},{i,1,7}]; P4 = ListPlot[B,PlotStyle->{RGBColor[1,0,0],Thickness[0.015]}]; P5 = Show[P1,P2,P4, PlotLabel->"P422 Ex 5 x^(2/3) + y^(2/3) = 1, distance=1/4", AspectRatio->Automatic]; Display["ex5x.ps",P5]; --------------------------------------------------------- Find the arc length of the star. 2/3 2/3 x + y = 1 / 2/3 x^(-1/3) + 2/3 y^(-1/3) y = 0 1/3 / y y = - -------- 1/3 x 2/3 2/3 / 2 x + y 1 1+ (y ) = -------------------- = ---------- 2/3 2/3 x x _ _ x=1 x=1 -1/3 | 2/3 | x=1 INT ds = INT x dx =| 3/2 x | = 3/2 x=0 x=0 |_ _| x=0 The total perimeter is 6. Page 444 Example 6: x= 1 1 INT ------ dx x=-1 2 x Page 444 Example 7 x=3 dx INT --------- x=0 2/3 (x-1) x=2 Pi Cos[x] INT ---------- dx x=0 1+Sin[x]