# How do you find the arc length of #y=ln(cos(x))# on the interval #[pi/6,pi/4]#?

##### 1 Answer

You can find the Arc Length of a function by first finding its derivative and plugging into the known formula:

#L = int_a^bsqrt(1 + (dy/dx)^2)dx#

**Process:**

With our function of

#dy/dx = -sinx/cosx# ,

which is equal to:

#-tanx# .

Plugging into our Arc Length formula, we have:

#L = int_a^b sqrt(1 + (-tanx)^2)dx# .

If we square the

#L = int_a^b sqrt(1 + tan^2(x))dx#

Since

#L = int_a^b sqrt(sec^2(x))dx# , which simplifies to#L = int_a^b secx dx#

Now we must remember that

#ln(secx + tanx)# from#pi/6# to#pi/4# , giving us:

#L = ln(2/sqrt2 + 1) - ln(2/sqrt3 + 1/sqrt3)#

#L = ln(sqrt2 + 1) - ln(sqrt3)#

If you remember that

#L = ln((sqrt2 + 1)/sqrt3)#

We can evaluate this for a decimal answer:

#L ~~ 0.332067...#