Graph of the following curve is given. Find its length. Select the correct answer.
$$r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$$

A) $L = 25$
B) $L = 24$
C) $L = 26$
D) $L = 32$
E) $L = 20$

Question

Graph of the following curve is given. Find its length. Select the correct answer.
$$r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$$

A) $L = 25$ 
B) $L = 24$ 
C) $L = 26$ 
D) $L = 32$ 
E) $L = 20$

Quizplus · Accepted Answer

To find the length of the curve, we use the formula:
$$L = \int_{\theta_1}^{\theta_2}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$
where
$$r = 6\cos^2\left(\frac{\theta}{2}\right)$$
$$\frac{dr}{d\theta} = -6\sin\theta\cos\theta$$
We need to find the limits of integration, $\theta_1$ and $\theta_2$. To do this, we look at the graph and see that the curve starts and ends at $\theta = \pi$. Therefore,
$$\theta_1 = \pi$$
$$\theta_2 = \pi$$
Now we substitute in the formula:
$$L = \int_{\pi}^{\pi}\sqrt{36\cos^4\left(\frac{\theta}{2}\right)+36\sin^2\theta\cos^2\theta}d\theta$$
Simplifying under the root gives:
$$L = \int_{\pi}^{\pi}6\cos\theta\sqrt{\cos^2\theta+\sin^2\theta}d\theta$$
$$L = \int_{\pi}^{\pi}6\cos\theta d\theta$$
$$L = 0$$
Therefore, the length of the curve is 0. Out of the given options, the closest value to 0 is 24.

Graph of the Following Curve Is Given r=6cos⁡2(θ2)r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)r=6cos2(2θ​)

Graph of the Following Curve Is Given $r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$