# What is the arclength of #f(x)=(1-3x)/(1+e^x)# on #x in [-1,0]#?

##### 1 Answer

#### Explanation:

Arc length formula of

Hence, substituting into the formula, we have

Arc Length =

Since

You could try to *simplify* this integral, but soon one will realise that this does not have a "nice" closed form

At this point, you can only approximate this integral using Riemann sums or infinite series, and the answer you get will be around

**Derivation of Arc Length Formula**

Given a function

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 x 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 be in terms of the x interval

Integrating both sides across the interval