Asked by sammyg
Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.)
y = ln (sin(x))
,[π/4,3π/4]
y = ln (sin(x))
,[π/4,3π/4]
Answers
Answered by
Steve
y = ln sinx
y' = 1/sinx * cosx = tanx
s = Int(sqrt(1+(y')^2)dx)[pi/4,3pi/4]
= Int(sqrt(1+tan^2(x))dx)[pi/4,3pi/4=
= Int(secx dx)[pi/4,3pi/4]
= ln|secx + tanx|[pi/4,3pi/4]
= ln|-1/√2 + 1| - ln|1/√2 + 1|
= ln|(1-√2/(1+√2)|
= ln|2√2-3|
= ln(3-2√2)
y' = 1/sinx * cosx = tanx
s = Int(sqrt(1+(y')^2)dx)[pi/4,3pi/4]
= Int(sqrt(1+tan^2(x))dx)[pi/4,3pi/4=
= Int(secx dx)[pi/4,3pi/4]
= ln|secx + tanx|[pi/4,3pi/4]
= ln|-1/√2 + 1| - ln|1/√2 + 1|
= ln|(1-√2/(1+√2)|
= ln|2√2-3|
= ln(3-2√2)