1336-Arclength.sagews

Arclength

Let's start by examining arclength.  To find arc length, you just add up a bunch of small parts of speed, $ds=|r'| dt$.

var('t')
r=vector([cos(t), sin(t)])

dr=diff(r, t)
dr

(-sin(t), cos(t))

speed=norm(dr)
speed

sqrt(abs(-sin(t))^2 + abs(cos(t))^2)

ds=speed
ds

sqrt(abs(-sin(t))^2 + abs(cos(t))^2)

ds.simplify_full().simplify_trig()

1

integrate(ds.simplify_full().simplify_trig(), t, 0, 2*pi)

2*pi

numerical_integral(ds,  0, 2*pi)

(6.2831853071795862, 6.9757369960172638e-14)

parametric_plot(list(r), (t, 0, 2*pi))

The same procedure works in 3d (wow, you gotta love these vectors; we can do computations in whatever dimension we need!)

var('t')
r=vector([cos(t), sin(t)])
dr=diff(r, t)
speed=norm(dr)
ds=speed
integrate(ds, t, 0, 2*pi)

pi