What is the arclength of #f(x)=(1-3x)/(1+e^x)# on #x in [-1,0]#?
Arc length formula of
Hence, substituting into the formula, we have
Arc Length =
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