Mathematical Immaturity

14.13 Exercises

1. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = a(1 - \cos t)\mathbf{i} + a(t - \sin t)\mathbf{j}, \quad 0 \leq t \leq 2\pi, \quad a > 0$$
2. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = e^t\cos t\mathbf{i} + e^t\sin t\mathbf{j}, \quad 0 \leq t \leq 2$$
3. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = a(\cos t + t\sin t)\mathbf{i} + a(\sin t - t\cos t)\mathbf{j}, \quad 0 \leq t \leq 2\pi, \quad a > 0$$
4. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = \frac{c^2}{a}\cos^3 t\mathbf{i} + \frac{c^2}{b}\sin^3 t\mathbf{j}, \quad 0 \leq t \leq 2\pi, \quad c^2 = a^2 - b^2, \quad 0 < b < a$$
5. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = a(\sinh t - t)\mathbf{i} + a(\cosh t - 1)\mathbf{j}, \quad 0 \leq t \leq T, \quad a > 0$$
6. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = \sin t\mathbf{i} + t\mathbf{j} + (1 - \cos t)\mathbf{k}, \quad 0 \leq t \leq 2\pi$$
7. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = t\mathbf{i} + 3t^2\mathbf{j} + 6t^3\mathbf{k}, \quad 0 \leq t \leq 2$$
8. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = t\mathbf{i} + \log(\sec t)\mathbf{j} + \log(\sec t + \tan t)\mathbf{k}, \quad 0 \leq t \leq \frac{1}{4}\pi$$
9. Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = a\cos \omega t\mathbf{i} + a\sin \omega t\mathbf{j} + b\omega t\mathbf{k}, \quad t_0 \leq t \leq t_1$$
10. Find an integral similar to that in (14.18) for the length of the graph of an equation of the form $x = g(y),$ where $g$ has a continuous derivative on an interval $[c, d].$
11. A curve has the equation $y^2 = x^3.$ Find the length of the arc joining $(1, -1)$ to $(1, 1).$
12. Two points $A$ and $B$ on a unit circle with center at $O$ determine a circular sector $AOB.$ Prove that the arc $AB$ has a length equal to twice the area of the sector.
13. Set up integrals for the lengths of the curves whose equations are: (a) $y = e^x,$ $0 \leq x \leq 1$ (b) $x = t + \log t,$ $y = t - \log t,$ $1 \leq t \leq e$ Show that the second length is $\sqrt{2}$ times the first one.
14. (a) Set up the integral which gives the length of the curve $y = c\cosh(x/c)$ from $x = 0$ to $x = a$ $(a > 0, c > 0).$ (b) Show that $c$ times the length of this curve is equal to the area of the region bounded by $y = c\cosh(x/c),$ the $x$-axis, the $y$-axis, and the line $x = a.$ (c) Evaluate this integral and find the length of the curve when $a = 2.$
15. Show that the length of the curve $y = \cosh x$ joining the points $(0, 1)$ and $(x, \cosh x)$ is $\sinh x$ if $x > 0.$
16. A nonnegative function $f$ has the property that its ordinate set over an arbitrary interval has an area proportional to the arc length of the graph above the interval. Find $f.$
17. Use the vector equation $\mathbf{r}(t) = a\sin t\mathbf{i} + b\cos t\mathbf{j},$ where $0 < b < a,$ to show that the circumference $L$ of an ellipse is given by the integral: $$L = 4a\int_0^{\pi/2}\sqrt{1 - e^2\sin^2 t}\,dt$$ where $e = \sqrt{a^2 - b^2}/a$ (The number $e$ is the eccentricity of the ellipse.) This is a special case of an integral of the form: $$E(k) = \int_0^{\pi/2}\sqrt{1 - k^2\sin^2 t}\,dt$$ called an elliptic integral of the second kind, where $0 \leq k < 1.$ The numbers $E(k)$ have been tabulated for various values of $k.$
18. If $0 < b < 4a,$ let $\mathbf{r}(t) = a(t - \sin t)\mathbf{i} + a(1 - \cos t)\mathbf{j} + b\sin \frac{t}{2}\mathbf{k}.$ Show that the length of the path traced out from $t = 0$ to $t = 2\pi$ is $8aE(k),$ where $E(k)$ has the meaning given in Exercise 17 and $k^2 = 1 - (b/4a)^2.$
19. A particle moves with position vector $$\mathbf{r}(t) = t\mathbf{A} + t^2\mathbf{B} + 2\left(\frac{2}{3}t\right)^{3/2}\mathbf{A} \times \mathbf{B}$$ where $\mathbf{A}$ and $\mathbf{B}$ are two fixed unit vectors making an angle of $\pi/3$ radians with each other. Compute the speed of the particle at time $t$ and find how long it takes for it to move a distance of 12 units of arc length from the initial position $\mathbf{r}(0).$
20. (a) When a circle rolls (without slipping) along a straight line, a point on the circumference traces out a curve called a cycloid. If the fixed line is the x-axis and if the tracing point $(x,y)$ is originally at the origin, show that when the circle rolls through an angle $\theta$ we have $$x = a(\theta - \sin \theta), \quad y = a(1 - \cos \theta)$$ where $a$ is the radius of the circle. These serve as parametric equations for the cycloid. (b) Referring to part (a), show that $dy/dx = \cot \frac{1}{2}\theta$ and deduce that the tangent line of the cycloid at $(x,y)$ makes an angle $\frac{1}{2}(\pi - \theta)$ with the x-axis. Make a sketch and show that the tangent line passes through the highest point on the circle.
21. Let $C$ be a curve described by two equivalent functions $X$ and $Y,$ where $Y(t) = X[u(t)]$ for $c \leq t \leq d.$ If the function $u$ which defines the change of parameter has a continuous derivative in $[c,d]$ prove that $$\int_{u(c)}^{u(d)} \|X'(u)\|\,du = \int_c^d \|Y'(t)\|\,dt$$ and deduce that the arc length of $C$ is invariant under such a change of parameter.
22. Consider the plane curve whose vector equation is $\mathbf{r}(t) = t\mathbf{i} + f(t)\mathbf{j},$ where $$f(t) = t\cos\left(\frac{\pi}{2t}\right) \text{ if } t \neq 0, \quad f(0) = 0$$ Consider the following partition of the interval $[0,1]:$ $$P = \left\{0, \frac{1}{2n}, \frac{1}{2n-1}, \ldots, \frac{1}{3}, \frac{1}{2}, 1\right\}$$ Show that the corresponding inscribed polygon $\pi(P)$ has length $$|\pi(P)| > 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2n}$$ and deduce that this curve is nonrectifiable.