<sagecell>

  1. The code below will find the surface area of a surface that lies below f(x,y) and above the curve r(t) in the xy plane.

f(x,y)=9-x^2-y^2 r(t)=(2*cos(t), 3*sin(t)) trange=(t,0,2*pi)

ds=r.diff(t).norm() dA=f(x=r(t)[0],y=r(t)[1])*ds(t)

  1. Let's make a function right off to evaluate an integral over this curve.
  2. We'll use a numerical integral, because sometimes we might not be able to get an exact integral.

def line_integral(integrand): 

    return RR(numerical_integral(integrand, trange[1], trange[2])[0])

A = line_integral(dA(t))

show(table([ [r"$f(x,y)$",f(x,y)], [r"$\vec r(t)$",r(t)], [r"$d\vec r(t)/dt$",r.diff(t)(t)], [r"$ds$",ds(t)], [r"$d\sigma=f ds$",dA(t)], [r"$Surface Area = \sigma =\int_C f ds$",A], ]))

  1. This section makes the plot of a the surface.

r1(t,u)=(r(t)[0], r(t)[1],u*f(x=r(t)[0],y=r(t)[1])) curve_options=dict(color='red',thickness=3) curves=parametric_plot(r1(t,0), trange,**curve_options)+parametric_plot(r1(t,1),trange,**curve_options) sheet=parametric_plot(r1,trange,(u,0,1)) show(curves+sheet)

  1. If the integral is doable, then this will give the integral.

html("If the integral can be computed exactly by Sage, then it will eventually show up below. Otherwise, you'll get a huge list of errors.") integrate(dA(t), trange)

</sagecell>