Find the length of the curve $r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }$ answer to two decimal places.
A)0.58
B)3.46
C)2.00
D)1.15
E)1.73

Question

Quizplus · Accepted Answer

The length $L$ of a curve given in polar coordinates $r = f(\theta)$ from $\theta = a$ to $\theta = b$ can be found using the formula $L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$. For $r = 2\sec(\theta)$, $\frac{dr}{d\theta} = 2\sec(\theta)\tan(\theta)$. Substituting $r$ and $\frac{dr}{d\theta}$ into the formula and integrating from $0$ to $\frac{\pi}{3}$, we find the length of the curve. The calculation involves trigonometric identities and integration techniques, leading to the correct answer of approximately 3.46.

Find the Length of the Curve r=2sec⁡θ over the interval 0≤θ≤π3. Round your r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }r=2secθ over the interval 0≤θ≤3π​. Round your

Find the Length of the Curve $r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }$