[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} \newcommand{\dilemma}[2] { \left\{ \begin{array}{ll} #1 \\ #2 \end{array} \right. } \newcommand{\trilemma}[3] { \left\{ \begin{array}{ll} #1 \\ #2 \\ #3 \end{array} \right. } [/math]

Draw the graph of [math]f[/math], and evaluate [math]\int_a^b f(x) \; dx[/math] in each of the following examples.

• [math]\trilemma{1 & \mbox{if $-\infty \lt x \leq 0 $},} {5 & \mbox{if $0 \lt x \lt 2$},} {3 & \mbox{if $2 \leq x \lt \infty $},}[/math] and [math][a,b] = [-3,3][/math].
• [math]f(x) = \dilemma{x^2&\mbox{if $-\infty \lt x \lt 0 $},} {2-x^2&\mbox{if $0\leq x \lt \infty$},}[/math] and [math][a,b] = [-2,2][/math].
• [math]f(x) = n \quad[/math] if [math]n \leq x \lt n+1[/math] where [math]n[/math] is any integer, and [math][a,b] = [0,5][/math].