8.2 Area of a Surface of Revolution

1-2. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.


1. , ; -axis.

 .


2. , ; -axis.

 .


3-7. Find the area of the surface obtained by rotating the curve about the -axis.


3. Find the area of the surface when goes to 0.   . .

http://matrix.skku.ac.kr/cal-lab/cal-8-2-3.html  

 

 

k,x=var('k,x');

@interact

def _(y = input_box(1/2*k*x^2, label="y="), kappa=slider(0,1,0.05,default=1,label='k')):

   html('$$AREA=%s$$'%( lim(integral(pi*2*1/2*k*x^2*sqrt(1+(k*x)^2),x,0,1),k=kappa) ))

   plot(1/2*kappa*x^2, x,0,1, color='purple', fill=true).show(aspect_ratio=1, xmin=0, xmax=1, ymin=0, ymax=0.5)


 

  Answer : 0.

 

4. Find the area of the surface when is a positive even integer.

        . .

http://matrix.skku.ac.kr/cal-lab/cal-8-2-4.html

 

k,x=var('k,x');

@interact

def _(y = input_box(sin(k*x), label="y="), kappa=slider(0,100,1,default=1,label=' k')):

   html('$$AREA=%s$$'%(

 lim(integral(2*pi*sin(k*x)*sqrt(1+diff(sin(k*x),x)),x,0,pi),k=kappa) ))

   plot(sin(kappa*x), x,0,pi, color='purple', fill=true).show(aspect_ratio=1, xmin=0, xmax=pi, ymin=-1, ymax=1)


 

  Answer : 0.


5. , .

.


6. , .

.


7. Find the area of the surface obtained by rotating the curve about the -axis.
     , .

http://matrix.skku.ac.kr/cal-lab/cal-8-2-7.html 

 

p1=plot(sqrt(1-(x)^2),(0,1),rgbcolor=hue(0.4))

p2=plot(-2*log(x)*x,(0,1),rgbcolor=hue(0.6))

show(p1+p2)


 

  The area of the surface half sphere is . Therefore, .


8-10. The given curve is rotated about the -axis. Find the area of the resulting surface.


8. , .

.


9. , .

.


10. , .

.


11. Find the area of the surface of the solid of revolution obtained by rotating about the -axis the circle

     , .

 The right semicircle and left semicircle equations are given by

          and , respectively.

          .


12. Use a CAS to find the area of the surface obtained by rotating the curve about the given axis. Use Simpson's Rule with .

          (a) , ; -axis.

http://matrix.skku.ac.kr/cal-lab/cal-8-2-12.html 

 

 

 

var('x');

y=x^5;

d=diff(y);

f(x)=y*(1+d^2)^(1/2);

n(2*pi*(0.1/3*(f(0)+4*f(0.1)+2*f(0.2)+4*f(0.3)+2*f(0.4)+4*f(0.5)+2*f(0.6)+4*f(0.7)+2*f(0.8)+4*f(0.9)+f(1))));

  3.37004769810923    

 (b) , ; -axis

 

 

var('y');

x=ln(y);

d=diff(x);

f(y)=x*(1+d^2)^(1/2);

k=4;

n(2*pi*(0.1*k/3*(f(1)+4*f(1+0.1*k)+2*f(1+0.2*k)+4*f(1+0.3*k)+2*f(1+0.4*k)+4*f(1+0.5*k)+2*f(1+0.6*k)+4*f(1+0.7*k)+2*f(1+0.8*k)+4*f(1+0.9*k)+f(1+1*k))));

  26.8537676835850