Suppose one were to construct a path on a sphere of radius r that
(like the earth) has designated poles and lines of latitude and
longitude.  The path is to begin on the equator at 0d longitude and
end at the north pole, with the condition that at each point on the
path, the north latitude and the west longitude are equal.  How long
is the path, in terms of r?

I have an approximate solution: about 1.9101r, which I got by using a
formula I derived (years ago) for determining the great-circle
distance between two points on a sphere given their respective
latitudes and longitudes, and applying it incrementally, Riemann-sum
style, to small segments of the path.  I imagine there's a way to
compute this directly--a path integral of some sort . . . but not
having taken calculus since 1975 I'm a little rusty!  Is there such an
integral?  Can the result be expressed exactly (say, with pi and
radicals)?

Just curious.

Thanks,
Eric

Let r be the radius, theta (henceforth t) be the longitude, and phi
(henceforth p) the latitude. The coordinate equations are

x = r cos t cos p
y = r sin t cos p
z = r sin p

We want p to go from 0 to pi/2 and also keep t = p. So we get our
coordinate as (r cos^2 p, r sin p cos p, r sin p).

The path length will be

Int {p=0..pi/2} sqrt((dx/dp)^2 + (dy/dp)^2 + (dz/dp)^2) dp
= Int {p=0..pi/2} sqrt([-2r cos p sin p]^2 
       + [r cos^2 p - r sin^2 p]^2 + [r cos p]^2) dp
= r. Int {p=0..pi/2} sqrt(4 cos^2 p sin^2 p + cos^4 p
       + sin^4 p - 2 cos^2 p sin^2 p + cos^2 p) dp
= r. Int {p=0..pi/2} sqrt([cos^2 p + sin^2 p]^2 + cos^2 p) dp
= r. Int {p=0..pi/2} sqrt(1 + cos^2 p) dp

Wolfram's online integrator gives the indefinite integral in terms of
an incomplete elliptic integral funtion as
sqrt(2).EllipticE(p, 1/2).
Wolfram's functions site also says that 
EllipticE(0, m) = 0
EllipticE(pi/2, m) = E(m) where the latter is a complete elliptic
integral funtion of the second kind, so we have

r. sqrt(2). E(1/2)

Unfortunately we can't do any better than that for an analytical answer.

Wolfram's functions site evaluates E(1/2) to 1.350643881 (10 sf), so
we get an answer of 1.910098894 r to 10 sf, which agrees nicely with
your empirical answer.

For future reference:
http://integrals.wolfram.com/
http://functions.wolfram.com/

Thanks, manuka, for your quick and excellent response.