Describe the given region in polar coordinates

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

Describing the region in polar coordinates:

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

• Lower portion:
  • $0\le \theta \le \frac{\pi }{6}0\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash frac\{\backslash pi\}\{6\}$​
  • $1\le r\le 6\sec \theta 1\; \backslash leq\; r\; \backslash leq\; 6\; \backslash sec\; \backslash theta$
    
• Upper portion:
  • $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\backslash frac\{\backslash pi\}\{6\}\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash frac\{\backslash pi\}\{2\}$​
  • $1\le r\le 6\csc \theta 1\; \backslash leq\; r\; \backslash leq\; 6\; \backslash csc\; \backslash theta$
    

So, the full description in polar coordinates is:

• Lower portion:
  • $0\le \theta \le \frac{\pi }{6}0\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash frac\{\backslash pi\}\{6\}$​
  • $1\le r\le 6\sec \theta 1\; \backslash leq\; r\; \backslash leq\; 6\; \backslash sec\; \backslash theta$
    
• Upper portion:
  • $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\backslash frac\{\backslash pi\}\{6\}\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash frac\{\backslash pi\}\{2\}$​
  • $1\le r\le 6\csc \theta 1\; \backslash leq\; r\; \backslash leq\; 6\; \backslash csc\; \backslash theta$