Exercises

1. A flat circular plate has the shape of the region . The plate is heated so that the temperature is given by

   

   Find the temperature and coordinates of the hottest and coldest points of the plate. Don't forget to check the boundary!

2. What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius R? You may assume that one side of the box lies in the planar base of the hemisphere.
3. Compute the following double integrals:
   1. 
   2. 
4. Find the mass and center of mass of the given laminas. The definition of mass and center of mass are defined on page 866 of the text.
   1. The region bounded by and where . Plot the region and indicate the centroid on your graph.
   2. The square with vertices (0,0), (0,a), (a,a) and (a,0) with . Sketch the region and indicate the centroid on your graph.
   3. The trapezoid with vertices and where are positive and . Assume that . Sketch the region and indicate the centroid on your graph.