Integration and Summation

Our goal here is to affirm that a double integral is a sum.

1. Consider a rectangular plate in the shape of the region bounded by the lines y=0, y=1, x=2, x=4. Suppose the density of the plate at the point (x,y) is . Compute the mass of this plate in two different ways as follows -
   (h (specified below). Then the mass of the plate is the sum of the mass of the subrectangles. Approximate the mass of each subrectangle by assuming that the density at any point on this subrectangle is the value of at the top left corner of the subrectangle.
   • Compute the approximate mass of the plate by hand assuming h=0.5.
   • Using Maple, compute the approximate mass of the plate, when h=0.1,0.01,0.001. Does your answer come close to the answer from (a).
2. Consider a plate in the form of an annulus bounded by the circles r=1 and r=2. Assume the density of the plate at the point (x,y) is . Find the mass of the plate by
   (h units apart i.e.  r=1, r=1+h, r=1+2h etc. and the lines , , , , etc. Here k is measured in radians. Approximate the mass of each subrectangle by assuming that the density on each subrectangle is constant and equals the value of at the ``top left'' corner of the subrectangle.
   • Compute the approximate mass by hand assuming h=0.5 and .
   • Using Maple, compute the approximate mass of the plate for the cases , .

 
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