Approximating a Double Integral with Cuboids

side divisions

3

4

5

6

7

8

9

10

method

lower sum

upper sum

midpoint rule

corner average

F(x,y) = 2-2sin

2

x

+

2

y



∫∫F(x,y)xy = 3.50968

volume = 3.22164

error = 0.288039

This Demonstration shows how the definite integral

1

∫

-1

1

∫

-1

2-2sin

2

x

+

2

y

xy

can be approximated using different numerical methods. The square region of integration is divided into subsquares according to the number of side divisions. The approximation is performed by finding the volume of the collection of cuboids over the subsquares. The heights of these cuboids are determined by the approximation method: the height for the lower sum is the infimum of

F

over the subsquare, for the upper sum the height is the supremum, for the midpoint rule the height is the value of

F

at the center of the subsquare, and for the corner average method it is the average of

F

at the four vertices of the subsquare. The error in approximation depends on the number of side divisions and the technique of approximation.