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   Anonymous

   2024-07-09T16:51:28-04:00Added an answer on July 9, 2024 at 4:51 pm

   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:
     • $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\backslash frac\{\backslash pi\}\{6\}\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash frac\{\backslash pi\}\{2\}$​
     • $1\le r\le 6\csc \theta 1\; \backslash leq\; r\; \backslash leq\; 6\; \backslash csc\; \backslash theta$
       

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