Path Lengths on a Sphere example.

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What does $\rho$ mean as to be measured along geodesics and more importantly how would i be able to parametrize this accordingly as being on the sphere's surface?

I know that I have to use the First fundamental Form, but struggle to use it as well............the path is in 3-D space (ie, great circle).

So how can I get around this? As a result of not being able to figure out the usage of $\rho$ I am unable to do part c). Finding the area.

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