How do you find the length of the curve #x=3t-t^3#, #y=3t^2#, where #0<=t<=sqrt(3)# ?
The answer is
The arclength of a parametric curve can be found using the formula:
It isn't very different from the arclength of a regular function:
We find the 2 derivatives:
And we substitute these into the integral:
Be aware that arclength usually has a difficult function to integrate. Most integrable functions look like the above where a binomial is squared and adding the two terms will flip the sign of the binomial.