[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]

Compute the upper and lower sums [math]U_\sigma[/math] and [math]L_\sigma[/math] in each of the following examples.

• [math]f(x) = \frac1x[/math], [math][a,b] = [1,4][/math], and [math]\sigma = \{1,2,3,4\}[/math].
• [math]f(x) = \frac x2[/math], [math][a,b] = [0,2][/math], and [math]\sigma = \{0, \frac13, \frac23, 1, \frac43, \frac53, 2\}[/math].
• [math]g(x) = x^2 + 1[/math], [math][a,b] = [0,1][/math], and [math]\sigma = \{x_0,x_1,x_2,x_3,x_4,x_5\}[/math], where [math]x_i = \frac i5, i = 0, \ldots, 5[/math].
• [math]g(x) = x^3[/math], [math][a,b] = [-1,1][/math], and [math]\sigma = \{-1, -\frac12, 0, \frac12, 1\}[/math].