222 - Topic 24 - Line Integral Lateral Surface

%auto ## Display and compute the area of the lateral surface between two surfaces ## corresponding to the (scalar) line integral ## John Travis ## Spring 2011 ## var('x,y,t,s') @interact(layout=dict(top=[['f','u'],['g','v']], left=[['a'],['b'],['in_3d'],['smoother']], bottom=[['xx','yy']])) def _(f=input_box(default=6-4*x^2-y^2*2/5,label='Top = $f(x,y) = $',width=30), g=input_box(default=-2+sin(x)+sin(y),label='Bottom = $g(x,y) = $',width=30), u=input_box(default=cos(t),label='   $ x = u(t) = $',width=20), v=input_box(default=2*sin(t),label='   $ y = v(t) = $',width=20), a=input_box(default=0,label='$a = $',width=10), b=input_box(default=3*pi/2,label='$b = $',width=10), xx = range_slider(-5, 5, 1, default=(-1,1), label='x view'), yy = range_slider(-5, 5, 1, default=(-2,2), label='y view'), in_3d = checkbox(default=true,label='3D'), smoother=checkbox(default=false), auto_update=true): ds = sqrt(derivative(u,t)^2+derivative(v,t)^2) # Set up the integrand to compute the line integral, making all attempts # to simplify the result so that it looks as nice as possible. A = (f(x=u,y=v)-g(x=u,y=v))*ds.simplify_trig().simplify() # It is not expected that Sage can actually perform the line integral calculation. # So, the result displayed may not be a numerical value as expected. # Creating a good but harder example that "works" is desirable. # line_integral = integrate(A,t,a,b) # html(r'<align=center size=+1>Lateral Surface Area = $ %s $ </font>'%latex(line_integral)) line_integral_approx = numerical_integral(A,a,b)[0] html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx)) # Plot the top function z = f(x,y) that is being integrated. G = plot3d(f,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2) G += plot3d(g,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2) # Add space curves on the surfaces "above" the domain curve (u(t),v(t)) G += parametric_plot3d([u,v,g(x=u,y=v)],(t,a,b),thickness=2,color='red') G += parametric_plot3d([u,v,f(x=u,y=v)],(t,a,b),thickness=2,color='red') k=0 if smoother: delw = 0.025 lat_thick = 3 else: delw = 0.10 lat_thick = 10 for w in (a,a+delw,..,b): G += parametric_plot3d([u(t=w),v(t=w),s*f(x=u(t=w),y=v(t=w))+(1-s)*g(x=u(t=w),y=v(t=w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9) if in_3d: show(G,stereo='redcyan',spin=true) else: show(G,perspective_depth=true,spin=true)