Justify, with a written explanation or a mathematical reasoning and with a sketch of at least two different cases, the following properties of integrals: abs(integrate from a to b of f(x)dx)...

Justify, with a written explanation or a mathematical reasoning and with a sketch of at least two different cases, the following properties of integrals:

abs(integrate from a to b of f(x)dx) is less than or equal to integrate from a to b of abs(f(x))dx

 

 

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

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One way to think about this:

You can take any function and write it in terms of its positive and negative parts:

`f=f_(+) - f_(-)`

To do this, when f is positive, then `f=f_(+)` and `f_(-) = 0`

when f is negative, then `f_(+) = 0` and `f_(-)=-f`

In pictures:

The black graph is `f` , the red one is `f_(+)` the blue one is `f_(-)`

Another example:

The nice thing about writing a function this way is:

`|f| = f_(+) + f_(-)`

 

So, to the question:

`int |f| dx = int (f_(+) + f_(-) )dx = int f_(+)dx + int f_(-) dx`

Remember, both `f_(+)` and `f_(-)` are nonnegative.  (positive or zero).  So both of those terms are nonnegative.

 

On the other hand:

`| int f dx| = |int (f_(+) - f_(-))dx| = |int f_(+) dx - int f_(-) dx|`

Again, `f_(+)` and `f_(-)` are nonnegative

So their integrals are nonnegative, and we are subtracting two positive numbers inside the absolute value sign.

 

Remember,

`|a-b| = a-b` or `|a-b|=b-a` , depending on whether a<b, a=b, a>b

 

So `|int f_(+) - int f_(-)| = int f_(+) - int f_(-) ` OR `int f_(-) - int f_(+)`

Either way, we are taking two positive quantities and subtracting one from the other.  But in the first case, we are taking those same two positive quantities and adding them together. 


So, if a and b are nonnegative, then

`a+b >= a-b` and `a+b >= b-a`

 

To recap, by splitting f into its positive and negative parts, we get that:

`int f_(+) + int f_(-) >= int f_(+) - int f_(-)`    AND

`int f_(+) + int f_(-) >= int f_(-) - int f_(+)`

The left hand side is another way to write `int |f|`

The right hand sides are the two possibilities for `| int f|`

 

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