Question

What is the arclength of `f(t) = (tlnt,t/lnt)` on `t in [1,e]`?

Answer 1

The arc length is infinite.

Explanation:

Notice that `lim_(trarr1^+)t/lnt=oo`, since `ln(1)=0`. This means that the arc length is infinite, as the function is asymptotic at `t=1`.

We can also note that if we attempt to apply the arc length formula `L=int_a^bsqrt((dx/dt)^2+(dy/dt)^2)dt`, the integral diverges.

Attached is a plot of the parametric graph for `tin(1,e]`: