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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:
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Linearity
| The integral of a linear combination of functions is the linear combination of the integrals of these functions. |
> #
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| | a f(x, y) + b g(x, y) dx dy =
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D
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a | | f(x, y) dx dy + b | | g(x, y) dx dy
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D D
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Example
> #
2 1
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| | 2
| | 2 x y - 3 y dx dy = -16
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0 -1
| and this is clearly equal to: |
> #
2 1 2 1
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| | | | 2
2 | | x y dx dy - 3 | | y dx dy = 2(0) - 3(16/3)
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0 -1 0 -1
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Monotonicity
| When, f(x,y) > g(x,y) for all the (x,y) in D then |
> #
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| | g dA < | | f dA
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Example
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On the unit square R=[0,1]x[0,1] we have,
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> 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
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| | | | 2
| | 2 x y dx dy < | | (x + y + 1) dx dy
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0 0 0 0
> #
1/2 < 25/6
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Area
| Area of bounded regions D, denoted here by A(D) can be computed by just integrating the constant function 1 over D. |
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A(D) = | | 1 dA
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D
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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 )
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| | 2
| | 1 dy dx = a Pi
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- a 2 2 1/2
- (a - x )
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Bounds
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From the previous three properties it follows that, if |
> #
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m A(D) < | | f dA < M A(D)
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D
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Example
| Over the rectangle R=[0,1]x[-1,2] |
> #
2 2
0 < x + y < 4
| hence, |
> #
2 1
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| | 2 2
0 < | | x + y dx dy = 4 < 4 (1-0)(2+1) = 12
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-1 0
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Additivity over D
| If we split the domain of integration D into two pieces, D1 and D2 then, |
> #
/ / / / / /
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| | f dA = | | f dA + | | g dA
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D = D1 U D2 D1 D2