# Properties of Double Integrals

 Integrals are limits of sums and as such they inherit most of their properties. For example if you multiply each term of a sum of terms by a number (the number 2 say) then the result of the sum is the same as if we multiply the original sum by 2. This is true for sums of a finite number of terms but it is still true for limits of these sums (when the limit exists) since the limits also have the property that the limit of a product is the product of the limits. Using familiar properties of sums and limits it is not difficult to show that integrals satisfy:

### Linearity

 The integral of a linear combination of functions is the linear combination of the integrals of these functions.

> #

```               /   /
|   |
|   |  a f(x, y) + b g(x, y) dx dy =
|   |
/   /
D
/   /                     /   /
|   |                     |   |
a  |   |  f(x, y) dx dy + b  |   |  g(x, y) dx dy
|   |                     |   |
/   /                     /   /
D                         D```

#### Example

> #

```                          2   1
/   /
|   |             2
|   |  2 x y - 3 y  dx dy = -16
|   |
/   /
0   -1```
 and this is clearly equal to:

> #

```                  2   1                 2   1
/   /                 /   /
|   |                 |   |   2
2  |   |  x y dx dy - 3  |   |  y  dx dy = 2(0) - 3(16/3)
|   |                 |   |
/   /                 /   /
0   -1                0   -1```

### Monotonicity

 When, f(x,y) > g(x,y) for all the (x,y) in D then

> #

```                         /   /             /   /
|   |             |   |
|   |  g dA   <   |   |  f dA
|   |             |   |
/   /             /   /```

#### Example

 On the unit square R=[0,1]x[0,1] we have, 2 x y < (x+y+1)2 and therefore,

> Int(Int(2*x*y,x=0..1),y=0..1) < Int(Int((x+y+1)^2,x=0..1),y=0..1);

```                  1   1                 1   1
/   /                 /   /
|   |                 |   |             2
|   |  2 x y dx dy <  |   |  (x + y + 1)  dx dy
|   |                 |   |
/   /                 /   /
0   0                 0   0```
> #
`                                   1/2 < 25/6`

### Area

 Area of bounded regions D, denoted here by A(D) can be computed by just integrating the constant function 1 over D.

> #

```                                      /   /
|   |
A(D) =  |   |  1 dA
|   |
/   /
D```

#### Example

 The area of a circle of radius a is given by,

> Int(Int(1,y=-sqrt(a^2-x^2)..sqrt(a^2-x^2)),x=-a..a) =
> int(int(1,y=-sqrt(a^2-x^2)..sqrt(a^2-x^2)),x=-a..a);

```                                2    2 1/2
a    (a  - x )
/         /
|         |                   2
|         |        1 dy dx = a  Pi
|         |
/         /
- a      2    2 1/2
- (a  - x )```

### Bounds

 From the previous three properties it follows that, if m < f(x,y) < M then,

> #

```                               /   /
|   |
m A(D) <  |   |  f dA  < M A(D)
|   |
/   /
D```

#### Example

 Over the rectangle R=[0,1]x[-1,2]

> #

```                                   2    2
0 <    x  + y  < 4```
 hence,

> #

```         2   1
/   /
|   |   2    2
0  <    |   |  x  + y  dx dy = 4  <  4 (1-0)(2+1) = 12
|   |
/   /
-1  0```

 If we split the domain of integration D into two pieces, D1 and D2 then,

> #

```                /   /             /   /             /   /
|   |             |   |             |   |
|   |  f dA    =  |   |  f dA    +  |   |  g dA
|   |             |   |             |   |
/   /             /   /             /   /
D = D1 U D2          D1               D2      ```

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>