exercise:04653989bc

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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)