Find the length of the curve.
-The parabolic segment $r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$
A) $\frac { 3 } { 2 } ( \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) )$
B) $6 \pi$
C) 3
D) $\frac { 3 } { 2 } ( \sqrt { 2 } - \ln ( \sqrt { 2 } - 1 ) )$

Question

Quizplus · Accepted Answer

The length of a curve in polar coordinates, given by $r = f(\theta)$, for $\theta$ in $[a, b]$, is calculated using the formula $L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$. For $r = \frac{3}{1 + \cos\theta}$, we differentiate to find $\frac{dr}{d\theta}$ and then integrate from $0$ to $\frac{\pi}{2}$. The correct answer involves a combination of trigonometric identities and integration techniques, leading to the specific form given in option A.

Find the Length of the Curve r=31+cos⁡θ,0≤θ≤π2r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }r=1+cosθ3​,0≤θ≤2π​

Find the Length of the Curve $r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$