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.ICN Textbook: Calculus Early Vectors Preliminary Edition By Stewart. Friday, March 7, 8.9 Improper Integration p516: 36,41,42,45,48,(52),54,(56),(70), (71) Main Idea: Bigger is Bigger; Smaller is Smaller. Key Words: Convergent, Divergent, Comparison test Goal: Learn estimation for integrals Previous Assignment Page 516 Problem 6 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x = -1 dx INT ----------- x = -Infinity (x-1)^(1/3) The only place where the function is not defined is at x=1 which is not in the interval [-Infinity, -1]. | 2/3 | x = -1 | (x-1) | | --------- | | 2/3 | x = a 3/2 ( (-2)^(2/3) - (a-1)^(2/3) ) Limit 3/2 ( (-2)^(2/3) - (a-1)^(2/3) ) = - Infinity. a-> -Infinity The interval diverges. --------------------------------------------------------------- Page 516 Problem 22 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x=Infinity dx INT -------- x=e x(ln x)^2 | | x=b | ln(x)^(-1) | | ------------- | | -1 | x=e -[ 1/ln(b) - 1 ] Limit b->Infinity = 1. ------------------------------------------- Mathematica does something weird. We try the obvious. In[1]:= Integrate[ 1/(x Log[x]^2),{x,E,Infinity}] 1 Integrate::idiv: Integral of --------- does not converge on {E, \[Infinity]}. 2 x Log[x] 1 Out[1]= Integrate[---------, {x, E, \[Infinity]}] 2 x Log[x] ---------------------------------------------------- We try taking the limit ourselves. In[1]:= Integrate[ 1/(x Log[x]^2),{x,E,b}] 1 Out[2]= 1 - ------ Log[b] 1 Limit 1- -------- = 1 b->Infinity Log[b] ----------------------------------------------------- We try numerical integration. In[3]:= NIntegrate[ 1/(x Log[x]^2),{x,E,Infinity}] Out[3]= 1. =================================================== Page 516 Problem 30 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x=9 dx INT ------------ x=1 (x-9)^(1/3) Watch out at x=9 since it goes off to infinity. x=b dx INT --------------- x=1 (x-9)^(1/3) | | x=b | (x-9)^(2/3) | | ------------ | | 2/3 | x = 1 3/2( (b-9)^(2/3) - (-8)^(2/3) ) 3/2( (b-9)^(2/3) -4 ) Limit 3/2( (b-9)^(2/3) -4 ) = -6. b->9 ------------------------------------------- Mathematica says: Remark: Avoid fractional exponents for negative numbers. They are not well defined. In particular 2/4 =/= 1/2 as fractional exponents. ____ 2/4 4/ 2 (-1) = \/ (-1) = 1 1/2 (-1) = i. -1/2 -1/2 So it is better to rewrite (x-9) to -(9-x) so that Mathematica only has to work with fractional exponents on positive numbers. Actually, What Mathematica will do is this. p/q b means to take the primitive qth root of a and raise it to the pth power. The qth roots of b lie on a circle in the complex plane and the nearest one above the x-axis is the primitive qth root. 1/3 (-1) = 0.5 + 0.866025 I 1/5 (-1) = 0.809017 + 0.587785 I Avoid this by keeping the base of fractional powers positive. Integrate[-(9-x)^(-1/3),{x,1,9}] = -6 ==================================================== New Material 2 x=Infinity -x Show that INT e dx is convergent. x=0 x=1 x=Infinity INT e^(-x^2) dx + INT e^(-x^2) dx x=0 x=1 this part is we will show this finite is finite also e^(-x^2) <= e^(-x) dx if x >= 1 x=Infinity x=Infinity | x=Infinity So INT e^(-x^2) dx <= INT e^(-x) dx -e^(-x) | = 1/e x=1 x=1 | x=1 ----------------------------------------------------------------------- x=Infinity We simply know that INT is finite. We do not even know that x=0 it is less than e since we do not know what the part from 0 to 1 is. But we can approximate it numerically once we know it exists. a = Plot[{E^(-x^2),E^(-x)},{x,0,3}]; b = Show[a, PlotLabel->"Page 516 Example 9",PlotRange->All]; Display["b.ps",%]; ---------------------------------------------------------- f[x_] := 1/x a = ParametricPlot3D[ {x, f[x] Cos[t], f[x] Sin[t]},{x,1,10},{t,0,2 Pi}]; b = Show[a,PlotLabel->"Gabriel Horn",PlotRange->All]; Display["horn.ps", b]; For the curve y = 1/x for x=1 to x=Infinity. (a) What is the area underneath the curve. (b) Rotate it around the x-axis. What is the volume. ans(a) Infinity ans(b) Pi -------------------------------------------------------- x=Infinity Sin[x]^2 INT ------------ dx x=1 x^2 ----------------------------------------------- x=Infinity dx INT ---------------- x=1 2x x+e ---------------------------------------------- x=Pi/2 dx INT ----------- x=0 x Sin[x] --------------------------------------------- x=Infinity Sqrt[1+Sqrt[x]] INT -------------- dx x=1 Sqrt[x]