Find the arc lengths of the following parametrized curves.

• [math]\dilemma{x = t+1,} {y = t^{\frac32}, & \mbox{from [/math](2,1)[math] to [/math](5,8)[math].}}[/math]
• [math]\dilemma{x = t^2,} {y = \frac23 (2t+1)^\frac32, & \mbox{from [/math]\left(x(0),y(0)\right) = (0, \frac23)[math] to to [/math]\left(x(4), y(4)\right) = (16,18)[math].}}[/math]
• [math]P(t) = (t^2, t^3)[/math], \quad from [math]P(0)[/math] to [math]P(2)[/math].
• [math]\dilemma{x(\theta) = a \cos^3\theta, & a \gt 0,} {y(\theta) = a \sin^3\theta, & \mbox{from [/math]\left(x(0), y(0)\right) = (a,0)[math] to [/math]\left(x(\frac{\pi}2), y(\frac{\pi}2)\right) = (0,a)[math].}}[/math]

A particle in motion in the plane has position equal to

at time [math]t[/math]. How far does the particle travel along its path from time [math]t=0[/math] to time [math]t=1[/math]?

Find the arc lengths of the graphs of each of the following functions [math]f[/math] between the points [math](a, f(a))[/math] and [math](b,f(b))[/math].

• [math]f(x) = x^\frac32[/math], [math]a=1[/math], and [math]b=4[/math].
• [math]f(x) = \frac23 (x^2+1)^\frac32[/math], [math]a=0[/math], and [math]b=2[/math].
• [math]f(x) = x^2[/math], [math]a=0[/math], and [math]b=\frac12[/math].
• [math]f(x) = \frac12(e^x + e^{-x})[/math], [math]a=-1[/math] and [math]b=1[/math].

Show that the circumference of an ellipse with the line segment joining [math](-a,0)[/math] and [math](a,0)[/math] as major axis and the line segment joining [math](0,-b)[/math] and [math](0,b)[/math] as minor axis is given by an integral

Evaluate the constants [math]K[/math] and [math]k[/math] in terms of [math]a[/math] and [math]b[/math]. (Do not attempt to evaluate the integral.)

• Let [math]g[/math] be a function which is continuously differentiable on the closed interval [math][c,d][/math]. Prove, as a corollary of Theorem \ref{thm 10.2.2}, that the arc length [math]{L_c}^d[/math] of the graph of the equation [math]x = g(y)[/math] between the points [math](g(c), c)[/math] and [math](g(d), d)[/math] is given by the formula
  [[math]] {L_c}^d = \int_c^d \sqrt{1+g^\prime(y)^2} \; dy . [[/math]]
• Find the arc length of the graph of the equation [math]x = \frac13(y^2+2)^\frac32[/math] between the point [math]\left( \frac{2\sqrt2}2, 0 \right)[/math] and the point [math](2\sqrt6, 2)[/math].
• Express as a definite integral the arc length of that part of the graph of the equation [math]x = 2y - y^2[/math] for which [math]x \geq 0[/math].

The coordinates of a particle in motion in the plane are given by

at time [math]t[/math]. What is the distance which the particle moves along its path of motion between the time [math]t=0[/math] and [math]t=2[/math]?

The same curve can be defined by more than one parametrization:

• lab{10.2.7a} Draw the curve defined parametrically by
  [[math]] \dilemma{x(t) = t,}{y(t) = t, & 0 \leq t \leq 1.} [[/math]]
• lab{10.2.7b} Draw the curve defined parametrically by
  [[math]] \dilemma{x(t) = \sin\pi t,}{y(t) = \sin\pi t, & 0 \leq t \leq 1.} [[/math]]
• Compute the arc lengths from [math]t=0[/math] to [math]t=1[/math] for the parametrizations in \ref{ex10.2.7a} and \ref{ex10.2.7b}.
• Give a geometric interpretation which explains the difference between the arc lengths obtained for the two parametrizations.

Let [math]P:[a,b] \goesto \R^2[/math] and [math]Q:[c,d]\goesto R^2[/math] be two parametrizations of the same curve [math]C[/math] such that all four coordinate functions are continuously differentiable. (A function is continuously differentiable if its derivative exists and is continuous at every number in its domain.) Then [math]P[/math] and [math]Q[/math] are called equivalent parametrizations of [math]C[/math] if there exists a continuously differentiable function [math]f[/math] with domain [math][a,b][/math] and range [math][c,d][/math] which has a continuously differentiable inverse function, and in addition satisfies (i) [math]f(a) = c[/math] and [math]f(b) = d[/math], (ii) [math]P(t) = Q(f(t)),[/math] for every [math]t[/math] in [math][a,b][/math].

• Using the Chain Rule and the Change of Variable Theorem for Definite Integrals (for the latter, see Theorem \ref{thm 4.6.6}), prove that equivalent parametrizations assign the same arc length to [math]C[/math].
• Show that
  [[math]] P(t) = (\cos t, \sin t), \quad 0 \leq t \leq \frac{\pi}2 , [[/math]]
  [[math]] Q(s) = \left( \frac{1-s^2}{1+s^2}, \frac{2s}{1+s^2}\right), \quad 0 \leq s \leq 1 , [[/math]]
  are equivalent parametrizations of the same curve [math]C[/math], and identify the curve.
• Show that
  [[math]] P(t) = (\cos t, \sin t), \quad 0 \leq t \leq 2\pi , [[/math]]
  and
  [[math]] Q(s) = (\cos 5t, \sin 5t), \quad 0 \leq t \leq 2\pi , [[/math]]
  are nonequivalent parametrizations of the circle.

Prove directly from the least upper bound definition that arc length is additive, i.e., that [math]{L_a}^b + {L_b}^c = {L_a}^c[/math].