Question

Find the length of the curve, L. y = ln(cos x), 0 ≤ x ≤ π/4

Answer 1

Length of curve, L=0.3828

Explanation:

We are given a curve
`y=\ln \cos(x)`

Length of arc L=
`\int ds`

Where,

`ds=√(1+y'^2)dx`

`\text{Where, }0\leq x \leq (\pi)/(4)`

`y=\ln \cos(x)`

Derivative of y

`y'=-\tan x`

Substitute y' into ds

`ds=√(1+\tan^2x)dx\Rightarrow \int |\sec x|dx`

`\text{Length of arc L}=\int ds`

`L=\int_(0)^(\pi/4) |\sec x|dx`

`L=\log|\tan x+\sec x|_(0)^(\pi/4)`

`L=\log|1+√(2)|\approx 0.3828`

Thus, Length of curve, L=0.3828