Show that the length of the curve y=log sec x between the points x=0 and x=π/3 is log (2+√3).
Solution:
The Length of the curve y=log sec x between the points x=0 and x=π/3 given a formula
𝜋/3
𝑠=∫
√1 + (𝑦′)2
0
𝑦 ′ = (log(sec(𝑥)))′ =
1
(sec(𝑥))′ = tan(𝑥)
sec(𝑥)
√1 + 𝑡𝑎𝑛2 𝑥 =
𝜋/3
𝑠=∫
0
1
cos⁡(𝑥)
𝑑𝑥
𝑥 𝜋 𝜋/3
𝜋 𝜋
𝜋
= 𝑙𝑜𝑔 |tan⁡( + )|
= 𝑙𝑜𝑔 |tan⁡( + )| − 𝑙𝑜𝑔 |tan ( )|
cos⁡(𝑥)
2 4 0
6 4
4
𝜋
𝜋
√3
tan (6 ) + tan⁡(4 )
+1
𝜋 𝜋
3
tan ( + ) =
=
= 2 + √3
𝜋
𝜋
6 4
1 − tan (6 ) ∗ tan⁡(4 ) 1 − √3
3
So
𝜋/3
𝑠=∫
0
𝑑𝑥
𝜋 𝜋
𝜋
== = 𝑙𝑜𝑔 |tan⁡( + )| − 𝑙𝑜𝑔 |tan ( )| = log⁡(2 + √3)
cos⁡(𝑥)
6 4
4