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.)

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.

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

Evaluation of double Integration

y C R δy x O,O δx

Evaluation of double Integration

y x (o,o) a

Xy=16 X=y Y=0 4 8

Y , (4,4) x