Approximate Volumes under a Surface using the Riemann Sum

 

Example:   (continued for part b)

 

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 }

 

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 b.

                             

 

 

 

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

 

                

 

 

Area

  

  Sample Point

     Coordinates

 

f(x,y)

  2xy

 

Individual Volumes

f(x,y) [ Area]

 

1

     1

             (0.5,0.5)

   0.5

       0.5

2

     1

             (1.5,0.5)

   1.5

       1.5

3

     1

             (2.5,0.5)

   2.5

       2.5

4

     1

             (0.5,1.5)

   1.5

       1.5

5

     1

             (1.5,1.5)

   4.5

       4.5

6

     1

             (2.5,1.5)

   7.5

       7.5

 

                                                        ∑ of individual volumes = total volume = 18    result

 


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