How do you find the arc length of the curve y = ln &#x2061;<!-- ⁡ --> cos &#x2061;<!--

madridomot

madridomot

Answered question

2022-05-30

How do you find the arc length of the curve y = ln cos x over the interval [ 0 , π / 3 ]?

Answer & Explanation

Miriam Payne

Miriam Payne

Beginner2022-05-31Added 10 answers

f ( x ) = tan ( x )
so we have
0 π 3 1 + tan 2 ( x ) d x = 2 \arc cot h ( 3 )
Note that
1 + tan 2 ( x ) = sin 2 ( x ) + cos 2 ( x ) cos 2 ( x ) = 1 cos 2 ( x )
Zeihergp

Zeihergp

Beginner2022-06-01Added 6 answers

y = ln ( cos x )
y = tan x
Arc length is given by:
L = 0 π 3 1 + tan 2 x d x
Simplify:
L = 0 π 3 sec x d x
Integrate directly:
L = [ ln | sec x + tan x | ] 0 π 3
Insert the limits of integration:
L = ln ( 2 + 3 )

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