(*
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

Tuesday, October 17:    Section 14.5

p1035:1,2,3,4,8
p1036:33
p1037:40,47

Main idea: A sheet is stretched until it fits the

curvature of the dome.

Key words: Region of the parameters. dr1,dr2,|dr1xdr2| 

Goal: Understand the surface area formula. 

Previous assignment.
*)
(* This is a hand-in-computer assignment *)
(* Have the computer draw a graph of the *)
(* Following 5 surfaces.                 *)


(* 1.  Rotate z = Sin[y] 0 <= y <= 2 Pi about the z-axis.  *)

p1 = ParametricPlot3D[{y Cos[t],y Sin[t],Sin[y]},{y,0,2 Pi},{t,0,2 Pi}];
q1 = ParametricPlot3D[{y Cos[t],y Sin[t],Sin[y]},{y,0,2 Pi},{t,0,  Pi}];

(* 2.  Rotate r = 2+Sin[t] 0 <= t <= 2 Pi about the x axis *)

p2 = ParametricPlot3D[{ (2+Sin[t]) Cos[t], (2+Sin[t])Sin[t] Cos[f],(2+Sin[t])Sin[t] Sin[f]},
{t,0,2 Pi},{f,0,2 Pi }]

q2 = ParametricPlot3D[{ (2+Sin[t]) Cos[t], (2+Sin[t])Sin[t] Cos[f],(2+Sin[t])Sin[t] Sin[f]},
{t,0,2 Pi},{f,-Pi/2,  Pi/2  }]

(* 3.  Rotate r = 2+Sin[t] 0 <= t <= 2 Pi about the y axis. *)

p3 = ParametricPlot3D[ { (2+Sin[t])Cos[t] Cos[f],(2+Sin[t])Sin[t],(2+Sin[t])Cos[t] Sin[f]},
{t,0,2 Pi},{f,0,2 Pi}]

q3 = ParametricPlot3D[ { (2+Sin[t])Cos[t] Cos[f],(2+Sin[t])Sin[t],(2+Sin[t])Cos[t] Sin[f]},
{t,   0 , Pi  },{f, 0,  Pi  }]


(* 4.  Rotate y=z 0 <= z <= 5 about the z axis.             *)

p4 = ParametricPlot3D[ {y Cos[t],y Sin[t],y},{t,0,2 Pi},{y,0,5}]


(* 5.  Draw the cylinder with axis the z axis, radius 3, bottom z=0, top z=5 *)

p5 = ParametricPlot3D[{3 Cos[t],3 Sin[t],z},{t,0,2 Pi},{z,0,5}]



(* Consider the surface (r Cos[t] , r Sin[t], r^3 ) 0<=t<=2 Pi, 0<=r<=1;   *)

ParametricPlot3D[{r Cos[t], r Sin[t], r^3},{r,0,1},{t,0,2 Pi}];



Find the surface area of the above.

Rr = (Cos[t],Sin[t],3 r^2);

Rt = (-r Sin[t], r Cos[t],0);

        |   i        j           k      |
RrxRt = |   Cos[t]  Sin[t]     3 r^2    |
        |-r Sin[t]   r Cos[t]    0      |


        |   i        j           k      |
RrxRt =r|   Cos[t]  Sin[t]     3 r^2    |
        |  -Sin[t]  Cos[t]       0      |
        

RrxRt = r(-3r^2 Cos[t], -3 r^2 Sin[t], 1)


|RrxRt|  = r Sqrt(9r^4 Cos[t]^2 + 9 r^4 Sin[t]^2 + 1)

|RrxRt|^2 = r(9r^4 + 1)


Integrate[ r Sqrt[9r^4 + 1],{t,0,2 Pi},{r,0,1}]


In[1]:= Integrate[ r Sqrt[9r^4 + 1],{t,0,2 Pi},{r,0,1}]

        Pi (3 Sqrt[10] + ArcSinh[3])
Out[1]= ----------------------------
                     6

In[2]:= N[%]

Out[2]= 5.91943