What is the arclength of #f(t) = (cos2t-sin2t,tan^2t)# on #t in [pi/12,(5pi)/12]#?
1 Answer
≈14.69 (no closed form)
Explanation:
To find the arc length
Of course, this will often be wildly inaccurate since most curves aren't exactly a straight line.
However, for a continuous function, the smaller time interval you choose, the more accurate the above approximation is (you can imagine zooming in on a function's graph: the more you zoom, the closer the function looks like a straight line)
Thus, at the limit when you shorten the time interval, you get
To manipulate it to suit the given parametric, we can adjust the equation such that
Since
Factorising a 2 out of the entire expression and simplifying, we have
To find the total arc length from
Mathematically, this is equivalent to integrating from
Now this expression has no closed form and is approximately 14.69