Question

find the length of x=ln cos y from y= -pie/4 to y= pie/4

Answer 1

find the length of x=ln cos y from y= -pie/4 to y= pie/4

The equation to use, for distance along a line, is

s = int [y1 to y2] √(1 + (x’)^2) dy

We have x = ln(cos(y))

x’ = -sin(y)/cos(y)

1 + (x’)^2 = 1 + sin^2(y)/cos^2(y) = (cos^2(y) + sin^2(y))/cos^2(y) = 1/cos^2(y)

√(1 + (x’)^2) = 1/cos(y) = sec(y)

Hence,

s = int [y1=-π/4 to y2=π/4] sec(y) dy

s = [ln(sec(y) + tan(y)] [-π/4 to π/4]

s = {ln(sec(π/4] + tan(π/4) – ln(sec(-π/4) + tan(-π/4)}

s = {ln(√2 + 1) – ln(√2 – 1)}

s = ln{(√2 + 1) / (√2 – 1)}

s = ln{(√2 + 1)^2 / (√2 – 1)(√2 + 1)} = ln{(2 + 2√2 + 1)^2 / (2 – 1)}

s = ln(2√2 + 3)