# What is the arclength of #f(t) = (cos2t-sin2t,tan^2t)# on #t in [pi/12,(5pi)/12]#?

##### 1 Answer

≈14.69 (no closed form)

#### Explanation:

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, you get

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

Since

Factorising a 2 out of the entire expression and simplifying, we have

To find the total arc length from

Mathematically, this is equivalent to integrating from

Now this expression has no closed form and is approximately 14.69