Solve the problem.
-Find the length of the curve $y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2$
A) $\ln ( \sqrt { 3 } + 2 )$
B) $\ln ( \sqrt { 3 } + 1 )$
C) $\ln ( \sqrt { 3 } )$
D) $1 - \ln ( \sqrt { 3 } + 2 )$

Question

Quizplus · Accepted Answer

The length of a curve $y = f(x)$ from $x = a$ to $x = b$ is given by the integral $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$. For $y = \ln(\csc(x))$, $\frac{dy}{dx} = -\cot(x)$. Thus, the integral becomes $\int_{\pi/6}^{\pi/2} \sqrt{1 + \cot^2(x)} dx = \int_{\pi/6}^{\pi/2} \csc(x) dx$, which evaluates to $\ln | \csc(x) + \cot(x) | $ from $\pi/6$ to $\pi/2$, resulting in $\ln(\sqrt{3} + 2)$.

Solve the Problem y=ln⁡(csc⁡x),π/6≤x≤π/2y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2y=ln(cscx),π/6≤x≤π/2

Solve the Problem $y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2$