What is the arclength of #f(t) = (t/sqrt(t^2+2),t/e^(t-2))# on #t in [-1,1]#?
1 Answer
Explanation:
Using the Arc Length Formula, we have
Since
Similarly,
Substituting into the formula, we have
This integral unfortunately cannot be expressed in terms of standard functions or constants, so the best you can get is either an infinite series or an approximation.
In this case, the integral evaluates to
For arclength type questions, please take note that the vast majority of the resultant integrals will, like this one, be unsolvable in terms of standard functions.
Explanation for Arc Length Formula
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 to be really (infinitesimally) small, you get
With
To manipulate it to suit the given parametric, we can adjust the equation such that
So for the arc length