[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]

For each of the following equations [math]r=f(\theta)[/math] and pairs of numbers [math]a[/math] and [math]b[/math], draw the region [math]R[/math] consisting of all points with polar coordinates [math](r,\theta)[/math] such that [math]a\leq\theta\leq b[/math] and [math]0\leq r\leq f(\theta)[/math]. Compute [math]\mbox{''area''}(R)[/math].

• [math]r=4\sin\theta[/math], [math]a=0[/math] and [math]b=\pi[/math].
• [math]r=\frac4{\sin\theta}[/math], [math]a=\frac\pi4[/math] and [math]b=\frac{3\pi}4[/math].
• [math]r=2\theta[/math], [math]a=\pi[/math] and [math]b=2\pi[/math].
• [math]r=\frac1{2\cos\theta+3\sin\theta}[/math], [math]a=0[/math] and [math]b=\frac\pi2[/math].
• [math]r=\sqrt{2\cos2\theta}[/math], [math]a=0[/math] and [math]b=\frac\pi4[/math]. (See Example)