##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)
# Let's make a function right off to evaluate an integral over this curve.
#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],
]))
#This section makes the plot of 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)
#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)
