# Describe the given region in polar coordinates

Describe the given region in polar coordinates.

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1. To describe the given region in polar coordinates, we need to convert the boundaries of the region from Cartesian coordinates to polar coordinates.

The region is bounded by:

1. The straight lines $x=4x = 4$ and $y=6y = 6$
2. The curve of the quarter circle with radius 6

### Converting boundaries to polar coordinates:

1. Quarter circle with radius 6:
• $r=6r = 6$
2. Line $x=4x = 4$

• In polar coordinates, $x=r\mathrm{cos}\theta =4x = r \cos \theta = 4$
• Therefore, $r=\frac{4}{\mathrm{cos}\theta }=4\mathrm{sec}\theta r = \frac\left\{4\right\}\left\{\cos \theta\right\} = 4 \sec \theta$
3. Line $y=6y = 6$y=6:
• In polar coordinates, $y=r\mathrm{sin}\theta =6y = r \sin \theta = 6$
• Therefore, $r=\frac{6}{\mathrm{sin}\theta }=6\mathrm{csc}\theta r = \frac\left\{6\right\}\left\{\sin \theta\right\} = 6 \csc \theta$

### Describing the region in polar coordinates:

The region is divided into two parts based on $\theta \theta$θ:

• Lower portion:
• $0\le \theta \le \frac{\pi }{6}0 \leq \theta \leq \frac\left\{\pi\right\}\left\{6\right\}$
• $1\le r\le 6\mathrm{sec}\theta 1 \leq r \leq 6 \sec \theta$
• Upper portion:
• $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\frac\left\{\pi\right\}\left\{6\right\} \leq \theta \leq \frac\left\{\pi\right\}\left\{2\right\}$
• $1\le r\le 6\mathrm{csc}\theta 1 \leq r \leq 6 \csc \theta$

So, the full description in polar coordinates is:

• Lower portion:
• $0\le \theta \le \frac{\pi }{6}0 \leq \theta \leq \frac\left\{\pi\right\}\left\{6\right\}$
• $1\le r\le 6\mathrm{sec}\theta 1 \leq r \leq 6 \sec \theta$
• Upper portion:
• $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\frac\left\{\pi\right\}\left\{6\right\} \leq \theta \leq \frac\left\{\pi\right\}\left\{2\right\}$
• $1\le r\le 6\mathrm{csc}\theta 1 \leq r \leq 6 \csc \theta$