Question

Let D be the region between the circles of radius 6 and radius 8 centered at the origin that lies in the third quadrant.
Express D in polar coordinates.
Select the correct answer below:
1) \(\displaystyle{D}={\left\lbrace{\left({r},\theta\right)}{\mid}{6}\leq{r}\leq{8},{0}\leq\theta\leq{\frac{{\pi}}{{{2}}}}\right\rbrace}\)
2) \(\displaystyle{D}={\left\lbrace\begin{array}{cc} {r}&\theta\end{array}\right)}{\mid}{6}\leq{r}\leq{8},\pi\leq\theta\leq{\frac{{{3}\pi}}{{{2}}}}\rbrace\)
3) \(\displaystyle{D}={\left\lbrace{\left({r},\theta\right)}{\mid}{0}\leq{r}\leq{8},\pi\leq\theta\leq{\frac{{{3}\pi}}{{{2}}}}\right\rbrace}\)
4) \(\displaystyle{D}={\left\lbrace{\left({r},\theta\right)}{\mid}{6}\leq{r}\leq{8},\pi\leq\theta\leq{2}\pi\right\rbrace}\)

Answer 1

Given:let D be the region between the circles of radius 6 and 8 centered at the origin that lies in the third quadrant express D in polar coordinates.
Here the region D is between the circles with radius r=6,r=8 centered at the origin
that is \(\displaystyle{6}\leq{r}\leq{8}\) and that lies in quadrant 3
So,
\(\displaystyle\pi\leq\theta\leq{\frac{{{3}\pi}}{{{2}}}}\)
Therefore,
\(\displaystyle{D}={\left\lbrace{\left({r},\therefore\right)}{\mid}{6}\leq{r}\leq{8},\pi\leq\therefore\leq{3}\pi{2}\right\rbrace}\)
Therefore the 2 nd option is the correct one