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Online Notes / Calculus I / Extras / Proof of Various Integral Properties
Calculus I

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Proofs of Derivative Applications Facts E-Book   Chapter   Section Area and Volume Formulas

 Proof of Various Integral Facts/Formulas/Properties

In this section we’ve got the proof of several of the properties we saw in the Integrals Chapter as well as a couple from the Applications of Integrals Chapter.

 

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Proof of :  where k is any number.

This is a very simple proof.  Suppose that  is an anti-derivative of , i.e. .  Then by the basic properties of derivatives we also have that,

 

and so  is an anti-derivative of , i.e. .  In other words,

                                           

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Proof of :  

This is also a very simple proof  Suppose that  is an anti-derivative of  and that  is an anti-derivative of .  So we have that  and .  Basic properties of derivatives also tell us that

 

and so  is an anti-derivative of  and  is an anti-derivative of .    In other words,

                          

Pf_Box

 

 

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Proof of :  

From the definition of the definite integral we have,

                             

 

and we also have,

                             

 

Therefore,

                                   

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Separator5

 

Proof of :  

From the definition of the definite integral we have,

                          

 Pf_Box

 

 

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Proof of :  

From the definition of the definite integral we have,

                                                

 

Remember that we can pull constants out of summations and out of limits.

Pf_Box

 

 

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Proof of :  

First we’ll prove the formula for “+”.  From the definition of the definite integral we have,

                              

 

To prove the formula for “-” we can either redo the above work with a minus sign instead of a plus sign or we can use the fact that we now know this is true with a plus and using the properties proved above as follows.

                                   

Pf_Box

 

 

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Proof of : , c is any number.

If we define  then from the definition of the definite integral we have,

                                   

Pf_Box

 

 

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Proof of : If  for  then .

From the definition of the definite integral we have,

                                   

 

Now, by assumption  and we also have  and so we know that

                                                            

So, from the basic properties of limits we then have,

                                                    

 

But the left side is exactly the definition of the integral and so we have,

                                               

Pf_Box

 

 

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Proof of : If  for  then .

Since we have  then we know that  on