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.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. Class dismisses at 8:30 today. Wednesday, February 23* 9.4 Inf. fcn. p423: 1-4,7,(6),(10),(16),23,24,29,33,34,(36), 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: February 21 9.3 Inf. Limits p419: 1,(10),(12),15,17,20,21,(24),(26) Page 419 Problem 10 x=Infinity x INT ----------- dx x=1 2 2 (1+x ) x= a x INT ----------- dx x=1 2 2 (1+x ) | 2 -1 | x=a | (1+x ) | | ----------- | | -1 (2) | x=1 -1/2 ----------- - ( -1/4 ) = 1/4 - (1/2)/(1+a^2) 2 1+a The limit as a-> Infinity is 1/4. The Area is 1/4. ---------------------------------------------------- f[x_] := x/(1+x^2)^2; a = Plot[f[x],{x,1,10}]; g[x_] = Integrate[f[x],x]; b = Plot[g[x]-g[1],{x,1,10},PlotStyle->{RGBColor[1,0,0]},AspectRatio->Automatic]; c = Show[a,b,PlotLabel->"Page 419 Problem 10 INT x/(1+x^2)^2"]; Display["10.ps",c]; ---------------------------------------------------------------- Page 419 Problem 12 x=Infinity ln(x) INT ---------- dx x=e x x=a ln(x) INT ---------- dx x=e x | 2 | x=a | ( ln(x) ) | | ---------- | | 2 | x=e 2 = 1/2 ( ln(a) ) - 1/2 Limit as a->Infinity ln(a) is infinity. thus the integral diverges to +Infinity. ---------------------------------------------------- f[x_] := Log[x]/x; a = Plot[f[x],{x,E,100}]; g[x_] = Integrate[f[x],x]; b = Plot[g[x]-g[E],{x,E,100},PlotStyle->{RGBColor[1,0,0]},AspectRatio->Automatic]; c = Show[a,b,PlotLabel->"Page 419 Problem 12 INT ln(x)/x"]; Display["12.ps",c]; ---------------------------------------------------------------- Page 419 Problem 24 x=Infinity -x INT e Sin[x] dx x=0 -x e -Cos[x] -x -e dx Sin[x] dx -x -x -x e Sin[x] = -e Cos[x] - INT e Cos[x] dx -x e Sin[x] -x -e dx Cos[x] dx -x -x -x -x INT e Sin[x] dx = -e Cos[x] - e Sin[x] - INT e Sin[x] dx -x -x INT e Sin[x] dx = 1/2 e ( -Cos[x] - Sin[x] ) / -x \/ 1/2 e ( Cos[x] + Sin[x] ) -x 1/2 e ( Sin[x] - Cos[x] ) x=Infinity -x | -x | x=Infinity INT e Sin[x] dx = 1/2 |e (-Cos[x]-Sin[x]) dx | x=0 | | x=0 -a = 1/2 e (-Cos[a]-Sin[a]) + 1/2 Limit as a-->Infinity is 1/2. ---------------------------------------------------- f[x_] := E^(-x) Sin[x]; a = Plot[f[x],{x,0,10}]; g[x_] = Integrate[f[x],x]; b = Plot[g[x]-g[0],{x,0,10},PlotStyle->{RGBColor[1,0,0]},AspectRatio->Automatic]; c = Show[a,b,PlotLabel->"Page 419 Problem 24 INT e^-x Sin[x]",PlotRange->All]; Display["24.ps",c]; ----------------------------------------------------------------------- Page 419 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 b 1 ln(--- ) - ln(---) = ln(2) b+1 2 ------------------------------------------------------------ f[x_] := 1/(x^2+x); a = Plot[f[x],{x,1,10}]; g[x_] = Integrate[f[x],x]; b = Plot[g[x]-g[1],{x,1,10},PlotStyle->{RGBColor[1,0,0]},AspectRatio->Automatic]; c = Show[a,b,PlotLabel->"Page 419 Problem 26 INT 1/(x^2+x)",PlotRange->All]; Display["26.ps",c]; ----------------------------------------------------------------------- ================================================================== New Material: x=2 dx Example 1: INT ------------ x=0 Sqrt[4-x^2] x=2 dx INT ------------ x=0 2 Sqrt[1-(x/2)^2] x=2 1/2 dx | | x=2 INT ---------------- = | ArcSin[x/2] | = Pi/2 x=0 Sqrt[1-(x/2)^2] | | x=0 --------------------------------------------------------- f[x_] := 1/Sqrt[4-x^2]; a = Plot[f[x],{x,0,2}]; b = Show[a,PlotLabel->"Page 421 Ex 1 y = 1/Sqrt[4-x^2]"]; Display["ex1.ps",b]; --------------------------------------------------------- x=16 1 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 ---------------------------------------------------------------- x=1 1 Example 3: INT ---- dx x=0 x | x=1 ln(x) | = ln(1) - ln(a) = + Infinity | x=a 2/3 2/3 Page 422 Example 5 x + y = 1 --------------------------------------------------------- f[x_] := (1-Abs[x]^(2/3))^(3/2); g[x_] := -(1-Abs[x]^(2/3))^(3/2); a = Plot[f[x],{x,-1,1}]; b = Plot[g[x],{x,-1,1}]; c = Show[a,b,PlotLabel->"Page 422 Ex 5 x^(2/3) + y^(2/3) = 1",AspectRatio->Automatic]; Display["ex5.ps",c]; --------------------------------------------------------- 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 y^(1/3) y' = - -------- x^(1/3) x^(2/3) + y^(2/3) 1 1+ (y')^2 = -------------------- = ---------- x^(2/3) x^(2/3) x=1 x=1 | x=1 INT ds = INT x^(-1/3) dx = 3/2 x^(2/3) | = 3/2 x=0 x=0 | x=0 The total perimeter is 6. Example 6: x=1 1 INT ------ dx x=-1 2 x Example 7 x=3 dx INT --------- x=0 2/3 (x-1) x=2 Pi Cos[x] INT ---------- dx x=0 (1+Sin[x]) ------------------------------------------------------------ f[x_] := Cos[x]/(1+Sin[x]); a = Plot[f[x],{x,0,3 Pi/2-0.1}]; b = Plot[f[x],{x,3 Pi/2+0.1,2 Pi}]; g[x_] = Integrate[f[x],x]; c = Plot[g[x]-g[0],{x,0,2 Pi},PlotStyle->{RGBColor[1,0,0]},AspectRatio->Automatic]; d = Show[a,b,c,PlotLabel->"Cos[x]/(1+Sin[x])",PlotRange->All]; Display["ex.ps",d]; -----------------------------------------------------------------------