Consider a parametrized curve x = f(u) for values of u between a and t. The pathlength fun...

Consider a parametrized curve x = f(u) for values of u between a and t. The pathlength function (or arclength function) based at a is given by

This function measures the distance an object travels along its path as the parameter u varies from a to t. Here we regard t as varying and a as fixed. (See Section 5.1 for a more thorough discussion.)

(a) Find the pathlength function for based at 0.

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(b) Find the pathlength function for the helix x = cos u i + sin u j + u k.

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(c) Find the pathlength function for the line x =(au + x0)i + (bu + y0)j + (cu + z0)k based at 0.