2.multiple - choice(5 points)\nin the context of finding the length of a curve defined in polar coordinates…

Explanation:

Step1: Recall arc - length formula in polar coordinates

The arc - length formula for a curve given in polar coordinates $r = r(\theta)$ is derived from the relationship between polar and Cartesian coordinates $x=r\cos\theta$ and $y = r\sin\theta$. First, we find $\frac{dx}{d\theta}=\frac{dr}{d\theta}\cos\theta - r\sin\theta$ and $\frac{dy}{d\theta}=\frac{dr}{d\theta}\sin\theta + r\cos\theta$. Then, by the arc - length formula $ds=\sqrt{(\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2}d\theta$.

Step2: Expand and simplify

[ \begin{align*} \left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2&=\left(\frac{dr}{d\theta}\cos\theta - r\sin\theta\right)^2+\left(\frac{dr}{d\theta}\sin\theta + r\cos\theta\right)^2\ &=\left(\frac{dr}{d\theta}\right)^2\cos^{2}\theta - 2r\frac{dr}{d\theta}\cos\theta\sin\theta+r^{2}\sin^{2}\theta+\left(\frac{dr}{d\theta}\right)^2\sin^{2}\theta+2r\frac{dr}{d\theta}\cos\theta\sin\theta + r^{2}\cos^{2}\theta\ &=\left(\frac{dr}{d\theta}\right)^2(\cos^{2}\theta+\sin^{2}\theta)+r^{2}(\sin^{2}\theta+\cos^{2}\theta)\ &=r^{2}+\left(\frac{dr}{d\theta}\right)^2 \end{align*} ] So, $ds = \sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^2}d\theta$.

Answer:

A. $ds=\sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^2}d\theta$