Mathematical Immaturity

14.19 Exercises

• A particle moves along a plane curve whose polar equation is $r = e^{c\theta},$ where $c$ is a constant and $\theta$ varies from 0 to $2\pi.$ (a) Make a sketch indicating the general shape of the curve for each of the following values of $c:$ $c = 0,$ $c = 1,$ $c = -1.$ (b) Let $L(c)$ denote the arc length of the curve and let $a(c)$ denote the area of the region swept out by the position vector as $\theta$ varies from 0 to $2\pi.$ Compute $L(c)$ and $a(c)$ in terms of $c.$
• (b) Recall that for a curve given by the polar equation $r = f(\theta),$ the arc length $s$ of the curve from $\theta = a$ to $\theta = b$ where $0 \leq a \leq b \leq a + 2\pi$ is $$ \begin{align*} \\ s &= \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta. \end{align*} $$ The area $a(R)$ swept out by the radius $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is $$ \begin{align*} \\ a(R) &= \frac{1}{2}\int_a^b f^2(\theta)\,d\theta. \end{align*} $$
• (a) The sketch can be found in the Visualization component. (b) If $r = e^{c\theta},$ then the arc length $L(c)$ from $\theta = 0$ to $\theta = 2\pi$ is: $$ \begin{align*} \\ L(c) &= \int_0^{2\pi} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta. \\ \\ &= \int_0^{2\pi} \sqrt{e^{2c\theta}\left(1 + c^2\right)}\,d\theta. \\ \\ &= \sqrt{1 + c^2}\int_0^{2\pi}e^{c\theta}\,d\theta \\ \\ &= \frac{\sqrt{1 + c^2}}{c}\,e^{c\theta}\,\Biggr|_0^{2\pi} \\ \\ &= \frac{\sqrt{1 + c^2}}{c}\left(e^{2c\pi} - 1\right) \end{align*} $$ for $c \neq 0.$ The area a(c) swept out by $r$ is: $$ \begin{align*} \\ a(c) &= \frac{1}{2}\int_0^{2\pi} f^2(\theta)\,d\theta \\ \\ &= \frac{1}{2}\int_0^{2\pi} e^{2c\theta}\,d\theta \\ \\ &= \frac{1}{4c}\,e^{2c\theta}\,\Biggr|_0^{2\pi} \\ \\ &= \frac{1}{4c}\left(e^{4c\pi} - 1\right). \end{align*} $$ for $c \neq 0.$ When $c = 0,$ $r = 1,$ and the curve is a unit circle. Thus, we have: $$ \begin{align*} \\ L(c) &= \frac{\sqrt{1 + c^2}}{c}\left(e^{2c\pi} - 1\right), \quad c \neq 0 \\ \\ L(0) &= 2\pi \\ \\ a(c) &= \frac{e^{4c\pi} - 1}{4c}, \quad c \neq 0 \\ \\ a(0) &= \pi \quad \blacksquare \end{align*} $$