- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.19 Exercises
- If a curve is given by a polar equation $r = f(\theta),$ where $a \leq \theta \leq b \leq a + 2\pi,$ prove that the arc length is $$ \begin{align*} \\ \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta \end{align*} $$
- Given the polar function $r = f(\theta),$ the position vector in terms of its radial component is $\mathbf{r} = r\mathbf{u}_r.$ Its derivative with respect to $\theta$ is then $$ \begin{align*} \\ \mathbf{v} = \frac{dr}{d\theta}\mathbf{u}_r + r\mathbf{u}_{\theta} \end{align*} $$ where $\mathbf{u}_{\theta} = \frac{d\mathbf{u}_{r}}{d\theta},$ and where $\mathbf{u}_{\theta}$ and $\mathbf{u}_{r}$ are perpendicular unit vectors.
- Let $\mathbf{r} = r\mathbf{u}_r$ be the radial position vector parameterized by $\theta,$ where $\mathbf{u}_r = (\cos\theta\,\mathbf{i} + \sin\theta\,\mathbf{j})$ is a unit vector with the same direction as $\mathbf{r}.$ Then, velocity is $$ \begin{align*} \\ \mathbf{v} &= \frac{dr}{d\theta}\mathbf{u}_r + r\mathbf{u}_{\theta} \end{align*} $$ where $\mathbf{u}_{\theta} = \frac{d\mathbf{u}_{r}}{d\theta} = (-\sin\theta\,\mathbf{i} + \cos\theta\,\mathbf{j})$ is a unit vector perpendicular to $\mathbf{u}_r.$ Speed $v(t)$ is $$ \begin{align*} \\ v(\theta) &= \sqrt{\mathbf{v} \cdot \mathbf{v}} \\ \\ &= \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}. \end{align*} $$ Then, arc length $s$ from $\theta = a$ to $\theta = b$ is: $$ \begin{align*} \\ s &= \int_a^b v(\theta)\,d\theta \\ \\ &= \int_a^b\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta \quad \blacksquare \end{align*} $$