Find the length of the curve over the given interval. $$r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$
A) $24 \pi$
B) $12 \pi$
C) $48 \pi$
D) $72 \pi$
E) $6 \pi$

Question

Quizplus · Accepted Answer

The length $L$ of a curve given in polar coordinates $r = f(\theta)$ for $\theta$ in $[a, b]$ can be found using the formula $L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$. For $r = 24\cos\theta$, $\frac{dr}{d\theta} = -24\sin\theta$. Substituting these into the formula gives $L = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{(24\cos\theta)^2 + (-24\sin\theta)^2} d\theta = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{576(\cos^2\theta + \sin^2\theta)} d\theta = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 24 d\theta = 24\theta\Big|_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = 24\pi$.

Find the Length of the Curve Over the Given Interval r=24cos⁡θ,−π2≤θ≤π2r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }r=24cosθ,−2π​≤θ≤2π​

Find the Length of the Curve Over the Given Interval $r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$