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