[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

• 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].