$r = 4(1 + \sin \theta)$
| θ |
0° |
30° |
60° |
90° |
120° |
150° |
180° |
210° |
240° |
270° |
300° |
330° |
360° |
| r |
4 |
6 |
7.46 |
8 |
7.46 |
6 |
4 |
2 |
0.54 |
0 |
0.54 |
2 |
4 |
$r = 4(1 + \sin \theta)$
$\dfrac{dr}{d\theta} = 4 \cos \theta$
$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$
$\displaystyle s = 2\int_{-\pi/2}^{\pi/2} \sqrt{16(1 + \sin \theta)^2 + 16 \cos^2 \theta} ~ d\theta$ †
$\displaystyle s = 2\int_{-\pi/2}^{\pi/2} 4\sqrt{(1 + \sin \theta)^2 + \cos^2 \theta} ~ d\theta$
$\displaystyle s = 8\int_{-\pi/2}^{\pi/2} \sqrt{(1 + 2\sin \theta + \sin^2 \theta) + \cos^2 \theta} ~ d\theta$
$\displaystyle s = 8\int_{-\pi/2}^{\pi/2} \sqrt{1 + 2\sin \theta + (\sin^2 \theta + \cos^2 \theta)} ~ d\theta$
$\displaystyle s = 8\int_{-\pi/2}^{\pi/2} \sqrt{1 + 2\sin \theta + 1} ~ d\theta$
$\displaystyle s = 8\int_{-\pi/2}^{\pi/2} \sqrt{2 + 2\sin \theta} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \sqrt{1 + \sin \theta} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \sqrt{(1 + \sin \theta) \times \dfrac{1 - \sin \theta}{1 - \sin \theta}} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \sqrt{\dfrac{1 - \sin^2 \theta}{1 - \sin \theta}} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \sqrt{\dfrac{\cos^2 \theta}{1 - \sin \theta}} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \dfrac{\cos \theta}{\sqrt{1 - \sin \theta}} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} \dfrac{\cos \theta}{(1 - \sin \theta)^{1/2}} ~ d\theta$
$\displaystyle s = 8\sqrt{2}\int_{-\pi/2}^{\pi/2} (1 - \sin \theta)^{-1/2} \cos \theta ~ d\theta$
$\displaystyle s = -8\sqrt{2}\int_{-\pi/2}^{\pi/2} (1 - \sin \theta)^{-1/2} (-\cos \theta ~ d\theta)$
$s = -8\sqrt{2} \left[ \dfrac{(1 - \sin \theta)^{1/2}}{1/2} \right]_{-\pi/2}^{\pi/2}$
$s = -16\sqrt{2} \left\{ [ 1 - \sin (\pi/2) ]^{1/2} - [ 1 - \sin (-\pi/2) ]^{1/2} \right\}$
$s = -16\sqrt{2} (0 - \sqrt{2})$
$s = 32 ~ \text{units}$ answer: A
Note:
You can input the line with † in your calculator to save your self from the integration process. Just make sure the angle is in RAD mode.
