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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,535

Here is a draft of my Introduction to Integration

I am still trying to work how best to present this, so any ideas or criticism is welcome.

(There may also be mistakes, let me know if so).

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi MathsIsFun

A good start. On the linked 'Integration Rules' page, I assume the integration by parts and substitution are still in preparation.

Because the fundamental theorem uses areas and some teachers gloss over 'reverse of differentiation' bit, I have met students who have two misconceptions about integration:

(i) They may think that integration is defined as the reverse of differentiation;

(ii) they may think you can only use it to find areas.

I like your approach as it makes it clear that (i) is not the case.

To avoid (ii) I wonder if you would consider re-wording this bit:

Integration is a method to find the area underneath the graph of a function like this:

Replace with eg. "Integration can be used in many ways such as to find volumes and to calculate, for example, centres of gravity, but it is easiest to start with finding the area underneath the graph of a function like this: "

This has reminded me that I said I would suggest some more for the double differentiation page. Sorry, this got shunted down my to-do list, but I'll try to get on with this soon.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Hi Bob

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,535

Hi Bob, good suggestion.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi Stefy

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

What! Why have I spent weeks trying to get 21122012 to do calculus properly.

These statements are not the same.

The fundamental theorem shows that if an area function F(x) can be constructed for the function f(x), then F'(x) = f(x).

So starting with F you can differentiate to get f.

But starting with f you do not necessarily get F. You might get F + C.

So they are not fully inverses operations to each other.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi MathsIsFun

Maybe that sentence is a bit long. So it might be better to split it into several shorter ones.

"Integration is a method for adding up infinesimal 'slices' to calculate a whole.

It can be used for many things eg. calculating a volume or working out a centre of gravity.

It is easiest to start with finding the area underneath the graph of a function like this:"

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi MIF;

Nice page. Very neat and colorful, easy to read. Glad you did not mention that most integrals can not even be done analytically, that scares people.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

bob bundy wrote:

hi Stefy

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

What! Why have I spent weeks trying to get 21122012 to do calculus properly.

These statements are not the same.

The fundamental theorem shows that if an area function F(x) can be constructed for the function f(x), then F'(x) = f(x).

So starting with F you can differentiate to get f.

But starting with f you do not necessarily get F. You might get F + C.

So they are not fully inverses operations to each other.

Bob

They are not inverse in the sense a function is inverse.

The Fundamental theory of Calculus states that the area below the graph of a function is equal to the difference of the antiderivative of the function at the two endpoints of the interval on whic you calculate the area...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,535

I use the word "reverse".

What do you all think of the tap/tank example?

And is it good that I gave an example from the Rules of Derivatives before Rules of Integration, or should I just go straight to Rules of Integration?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

The tap/tank example is a great analogy for integration! Big stuff made out of smaller stuff- it's all Calculus!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi MIF;

You can leave the page the way it is. Unless someone else has found something... Post it!

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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