# 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 *really* small changes in

To manipulate it to suit the given parametric, we can adjust the equation such that

So for the arc length