Instructor Hentzel
Office Phone: 515-294-8141
E-mail:   hentzel@iastate.edu
Math Department Fax:      515-294-5454
http://orion.math.iastate.edu/hentzel/class.166.05


Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. 


January 31     8.1  Substitution   

p375: 1-10, (28),30,(32),49,(50), 55-58,(68)

Main Idea:  Review of integration. 

Key Words: Completing the square, Substitution 

Goal: Learn to recognize the integral pieces. 


--------------------------------------------------------
Previous assignment:           

January 24     6.6  Mo Cent 

p310:1,(2),9,(12),(14),15,17-20,(23),24,28,29,

Page 310 Problem  2

John and Mary, weighing 180 and 110 pounds, respectively,

sit at opposite ends of a 12-foot teeter board with the

fulcrum in the middle.  Where should their 80 pound son

Tom sit in order for the board to balance.


John               .    Tom           Mary
180                     80             110
-6                 0    x                6


180(-6) + 80x + 110 6 = 0

x = 21/4

---------------------------------------------
Page 310 Problem 12
Find the centroid of the region bounded by the curves.

y = 1/2 (x^2-10)   y=0  between x=-2 and x=2

_
x  = 0 by symmetry.

        x= 2
    =    INT  1/2(x^2-10) (1/4) (x^2-10) dx   
        x=-2
_                                                  574/15
y    = ------------------------------------    =   -------- =    -286/130 = -2.20769
                                                    -52/3
        x= 2
    =    INT  1/2(x^2-10) dx   
        x=-2




Integrate[1/2 (x^2-10) 1/4 (x^2-10),{x,-2,2}]  = 574/15
Integrate[1/2 (x^2-10) ,{x,-2,2}]  = -52/3 

a = Plot[1/2(x^2-10),{x,-2,2}];
b = ListPlot[{{0,-286/130}},PlotStyle->{RGBColor[1,0,0]}];
c = ParametricPlot[{-2,y},{y,-3,0}];
d = ParametricPlot[{ 2,y},{y,-3,0}];
e = Plot[0,{x,-2,2}];
f = Show[a,b,c,d,e,PlotLabel->"Page 310 Problem 12",AspectRatio->Automatic];
Display["12.ps",f];




Page 310 Problem 14

y = x^2    y = x+3


x^2 = x+3        

x^2 - x - 3 = 0

    1 +/- Sqrt[13]
x = --------------
          2

     x=stop
M  =  INT      x^2 - x - 3 dx  
     x=start


     x=stop
My =  INT      (x^2 - x - 3) (1/2) (x^2 + x + 3) dx  
     x=start



     x=stop
Mx =  INT     x( x^2 - x - 3) dx  
     x=start

Mx/M = 11/5

My/M = 1/2



         -143 Sqrt[13]
Mx     = -------------
              30


         -13 Sqrt[13]
My     = ------------
              12


         -13 Sqrt[13]
M      = ------------
              6

start = (1-Sqrt[13])/2;
stop  = (1+Sqrt[13])/2; 
m = Integrate[ x^2 - x - 3,{x,start,stop}];
mx = Integrate[(x^2 - x -3) (1/2) (x^2 + x + 3),{x,start,stop}];
my = Integrate[x(x^2-x-3),{x,start,stop}];
Print[m," ",mx," ",my];



a = Plot[{ x^2,x+3},{x,start,stop}];
b = ListPlot[{{1/2,11/5}},PlotStyle->{RGBColor[1,0,0]}];
c = Show[a,b,AspectRatio->Automatic,PlotLabel->"Page 310 Problem 14"];
Display["14.ps",c];


Page 310 Problem 23

(9/16,31/16)






=============================================================

New Material

Example 1 page 372

                   x
            INT------------- dx
                     2  2
                Cos[x  ]  




ans = 1/2 Tan[x^2]^2 + C     <=== Book answer is wrong


Example 2    

                         3
            INT    ------------ dx
                            2
                   Sqrt[5-9x  ]



ans = ArcSin(3x/Sqrt[5] + C




Example 3   

                       1/x
                    6 e
             INT   ------  dx
                    x^2



ans =  -6 e^(1/x)


Example 4                 x 
                         e  
               INT  ---------------  dx
                           2x
                      4+9 e



ans = 1/6 ArcTan[ 3e^x /2) + C



Example 5
                     3        4
                INT x   Sqrt[x  +11] dx




ans = 1/6 (x^4+11)^(3/2) + C



Example 6  
                         Tan[t] 
                        a
                INT  ----------  dt
                              2
                        Cos[t]  

             Tan[t]
            a
ans =  ----------
          ln(a)



Example 7
                          7
               INT ---------------- dx
                       2
                      x -6x+25


ans =  7/4 ArcTan[ (x-3)/4 ] + C



Example 8
                       2
                 INT (x  -x)/(x+1) dx





ans = x^2/2 - 2x + 2 ln|x+1| + C



Example 9

                x=5           2
                INT   t Sqrt[t -4] dt
                x=2


ans = 1/3 21^(3/2)