1. Double Integration
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis.
The double integral of a positive function of two variables represents the volume of the region between the surface defined by the function z = ƒ(x, y)) (on the three dimensional Cartesian plane and the plane which contains its domain.)

2. Note : 1: If z = ƒ(x, y)) is a pure constant number then the double
Note : 1: If z = ƒ(x, y)) is a pure constant number then the double integration an area
2: The same volume can be obtained via the triple integral— the integral of a function in three variables of the constant function ƒ(x, y, z) = C over the above- mentioned region between the surface and the plane.

3. 6: Center of Mass and Moment of Inertia 7: Surface Area
Applications
Double integrals arise in a number of areas
of science and engineering, including computations of
1: Area of a 2D region
2: Volume
3: Mass of 2D plates
4: Force on a 2D plate
5: Average of a function
6: Center of Mass and Moment of Inertia
7: Surface Area

4. Evaluation of double Integration

5. y C R δy x
O,O δx

6. Evaluation of double Integration

14. y x
(o,o) a

16. Xy=16
X=y
Y=0 4 8

18. Y ,
(4,4) x