Approximate Volumes under a Surface using the Riemann Sum

 

Example:   Estimate the volume of a solid S that lies below the surface, f(x,y) = 2xy

and above the rectangle {(x,y), | 0  ≤  x  ≤ 3,  0  ≤  y  ≤ 2 }

 

a.  Use a Riemann sum with m = 3 and n = 2 and take the sample point to be the upper

     right corner of each rectangle.

 

b.  Use the midpoint rule to estimate the volume of the solid as in part a.

 

 

Strategy:  First show the rectangular grid and sample points for part a.

         

 

 

 

Next set up table to calculate the double Riemann Sum for part a.

 

                                  

 

 

Area

    Sample Point

     Coordinates

 
f(x*,y*)

Individual Volumes

f(x,y)][Area]

1

     1

             (1,1)

   2

           2

2

     1

             (2,1)

   4

           4

3

     1

             (3,1)

   6

           6

4

     1

             (1,2)

   4

           4

5

     1

             (2,2)

   8

           8

6

     1

             (3,2)

  12

         12

 

                                                        ∑ of individual volumes = total volume = 36    result

 

 

Click here to continue with part b of this example.

 


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