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CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265 <##### NEW ADDRESS ####

Thursday, September  28:  Review.

1.  Find the volume above z = x^2+y^2 which is below the plane  z = 4;

Integrate[ r,{t,0,2 Pi},{r,0,2},{z,r^2,4}];
ans = 8 Pi.

2.  Find the surface area of the surface z = x^2 + y^2 below the plane z = 4.

Integrate[ r Sqrt[1+ 4 r^2],{t,0,2 Pi},{r,0,2}] 
 
        (-1 + 17 Sqrt[17]) Pi
ans   = ---------------------
                  6

3.  Convert the following into Cylindrical coordinates.  Do not integrate.

       2    +Sqrt[4-x^2]   Sqrt[16-x^2-y^2] 
      INT   INT            INT               Sqrt[x^2+y^2] dzdydx.
       0   -Sqrt[4-x^2]     0 

Mathematica would only integrate the original problem numerically giving 30.8723

Integrate[r^2,{t,-Pi/2,Pi/2},{r,0,2},{z,0,Sqrt[16-r^2]}]

                         16 Pi
Out[7]= Pi (-4 Sqrt[3] + -----)
                           3
4.  Convert the following into Spherical coordinates.  Do not integrate.

       1    Sqrt[1-x^2]   Sqrt[1-x^2-y^2]
      INT   INT            INT             Sqrt[x^2+y^2+z^2] dz dy dx
       0     0              0

Integrate[Sqrt[x^2+y^2+z^2],{x,0,1},{y,0,Sqrt[1-x^2]},{z,0,Sqrt[1-x^2-y^2]}]

Numerically integrating gives 0.392699.

Integrate[p^3 Sin[f],{t, 0,Pi/2},{f,0,Pi/2},{p,0,1}]

        Pi
Out[1]= --
        8












5.  Set up the Integral for the moment about the xy plane  of the portion of 

the cone z = Sqrt[x^2 + y^2]  below z = h in Spherical coordinates. 


Integrate[p^3 Sin[f] Cos[f],{t,0,2 Pi},{f,0,Pi/4},{p,0,h Sec[f]}];

         4
        h  Pi
Out[2]= -----
          4

6.  Set up the Integral for the volume of the portion of the sphere x^2+y^2+z^2 = a^2

above z = b in cylindrical coordinates.

Integrate[r,{t,0,2 Pi},{r,0, Sqrt[a^2-b^2]},{z,b,Sqrt[a^2-r^2]}]

            2 3/2                            2 3/2
        2 (a )    Pi    2         3      2 (b )    Pi
Out[1]= ------------ - a  b Pi + b  Pi - ------------
             3                                3

7.  Set up the Integral for the surface area of the portion of the sphere

x^2 + y^2 + z^2 = a^2 above z = b using polar coordinates.

Integrate[a r/Sqrt[a^2 - r^2],{t,0,2 Pi},{r,0,Sqrt[a^2-b^2]}]

                   2            2
Out[1]= 2 (a Sqrt[a ] - a Sqrt[b ]) Pi

8.  Set up the integral for Iz for the sno-cone which is the

intersection of the Sphere x^2 + y^2 + z^2 = a^2 and the cone

z^2 = 2(x^2 + y^2) in spherical coordinates.

Integrate[p^4 Sin[f]^3,{t,0,2 Pi},{f,0,ArcTan[1/Sqrt[2]]},{p,0,a}]

                      5                               1
                 5   a  (-3 Sqrt[6] + Cos[3 ArcTan[-------]])
              2 a                                  Sqrt[2]
Out[1]= 2 Pi (---- + ----------------------------------------)
               15                       60

In[2]:= N[%]

                  5
Out[2]= 0.039727 a


9.   Set the integral for the volume inside the sphere x^2+y^2+z^2 = a^2

and above   z = x^2 + y^2 in Cylindrical coordinates. 

10.  Give a geometrical interpretation of the triple integral.


        2 Pi     Pi   6 Sin[phi]
        INT     INT    INT        p^2 Sin[f] d(rho) d(phi) d(theta)
         0       0      0